5 PO), 


MATHEMATICS Liprapy 
THE UNIVERSITY 


OF ILLINOIS 
LIBRARY 


The 
Frank Hall collection 
of arithmetics, 
presented by Professor 
H. L. Rietz of the 
University of Iowa. 


atO-FL $13 
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joe 


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ees BS 
ESSENTIALS OF ARITHMETIC 
ORAL AND- WRITTEN 


Book II 


FOR UPPER GRADES 


GORDON A.-SOUTHWORTH 


SUPERINTENDENT OF SCHOOLS, SOMERVILLE, MASSACHUSETTS 


LEACH, SHEWELL, AND SANBORN 
BOSTON NEW YORK CHICAGO 


THE ESSENTIALS OF ARITHMETIC. Book I. For 
Lower Grades. 


THE ESSENTIALS OF ARITHMETIC. Book II. For 
Higher Grades. - 


Both books are published with and without answers. 


—— Ol oes 
¢ 


Key to Book //. for use of Teachers only. 


Copyrienst, 1895, By GORDON A. SOUTHWORTH. 


Norwood JBress ; 
J. S. Cushing & Co.— Berwick & Smith. 
Norwood, Mass., U.S.A. 


— aw 5138 


a MATHEMATICS LIBRARY 


PREFACE. 


Boox II. of the present series follows its predecessor after a 
considerable interval of time. It is for upper grammar grades, or 
for all grades above the primary when but one text-book in arith- 
metic is required. The manner of treating elementary subjects pro- 
vides for this double adaptation. 

Much that has been deemed not to belong among the “ essentials,” 
though commonly found in arithmetics, has been omitted, or left 
accessible in a subordinate form in the appendix. The order of 
presentation is in the main the usual one, though previous acquain- 
tance with the rudiments of a subject has often been assumed, and 
some subjects have been introduced in a preparatory way a few steps 
in advance of the full and formal treatment, which thus becomes 
far easier to comprehend. 

It is of high importance to be quick with figures, and long practice 
is needed: exercises specified as oral, written, for dictation, ete., 
are accordingly given in abundance, alternately upon the subject in 
hand, or as constantly recurring reviews. 

But the methods suggested call for effort and study, and look to 
the mathematical training of older children in something more than 
mechanical figuring by imitation. What ought to be perceived or 
discovered by thinking and reasoning is not first stated outright in 
print, though often led up to by stimulating questions. Such teach- 
ing should develop habits of correct and ready expression, with 
intelligent and permanent grasp of simple principles and processes. 

As to division of time between solving problems and analyzing 
them, the teachers must decide; but it has been shown that princi- 


iii 


404022 


Hh PREFACE. 


ples and methods cannot be securely fixed by mere repetition, or the 
working of many examples. Fewer problems, if solved indepen- 
dently and logically analyzed, will do most toward attaining the 
highest purpose of arithmetical work. 

Many a principle is made conspicuous upon the page; definitions 
are collected in five groups, and arranged alphabetically for refer- 
ence; set rules are given only as a summary, also for convenient 
reference, in the appendix. 

An introduction to the study of algebra is included in the 
appendix; and throughout the book letters are conveniently used to 
represent unknown quantities. 

The contents of each section are given in side headings, to which 
the following index furnishes a complete guide. 


JUNE, 1895. 


INDEX. 


FULL-FACED FIGURES REFER TO THE APPENDIX. 


Cylinder, 
Decimals, 
Decimal system, 
Definitions, 


PAGE 


188, 224 


3-5, 90-103, 2 


2 


7, 46, 110, 147, 234 


Denominate numbers, 8, 43, 70, 81, 85, 
86, 100, 105, 111-115 


PAGE 

Addition, of integers, 10-12 
Of fractions, 59 
Of decimals, 95 
Accounts, 18, 84 
Angles and arcs, 115 
Annual interest, 14 
Algebra, Appendix ii. 
Average of accounts, 19 
Bank check, 206 
Bank discount, 193-199 
Of interest-bearing notes, 198 
Rule, 6 
Bills, 40 
Bills of exchange, 207, 208 
Bonds, 201 


Business forms, 18, 40, 180, 190, 198, 
199, 201, 206, 207, 208, 16 


Cancellation, 52, 66, 68 
Cash account, 18 
Check, 206 
Circles, 127-133 
Comparison of numbers, 47, 76, 78 
Complex fractions, 73 

Decimals, 91 
Commission, 175-178 
Compound interest, 191, 6, 14 
Cones, 224-226 

Frustum, 4 
Cube root, 7,19 
Customs, 188 


Difference between dates, 165 
Discount, bank, 193-198 
Trade, 170 
Successive, 171 
True, 204 
Division, of integers, 24-29 
Of fractions, 70-74 
Of decimals, 98 
Duties, 188 
Equations, 37 
Equation of payments, 17 
Exact interest, 169, 6 
Exchange, domestic, 205 
Foreign, 209 
Factoring, 19, 58 
Fractions, common, 3, 4, 54-90, 2 
Complex, 73 
Decimal, 90-103 
Changes in form of, 54-60 
Added and subtracted, 68, 64 
Multiplied, 65-70 
Divided, 70-74 
Practice Table, 70, 74 
Greatest common divisor, bed 


“i INDEX. 


PAGE 

Insurance, 173, 174 
Interest, general method, 106-109 
Bankers’ method, 160-1638 
One dollar method, 163-165 
Choice of methods, 166, 168 
Compound, 191 
Exact, 169 
Problems in, 204 
Rules, 5 
Legal rates, 15 
Leap years, 8 
Least common multiple, 60, 1, 8 
Literal quantities, 36, Ap. ii. 


Measurements of — 

Arcs and Angles, 115; Circles, 127- 
133; Cones, 224; Cylinders, 138, 
224; Floors, 119; Frustums, 4; 
Hypotenuse, 222; Land, 121, 9; 
Lines, 111, 222; Lumber, 135; 
Pyramids, 225; Prisms, 135-136 ; 
Rectangles, 117 ; Rhomboids, 125; 
Roofs, 120; Rules for, 3, 4; 
Spheres, 228; Trapeziums, 127; 
Trapezoids, 124; Triangles, 126; 


Wood, 154. 
Metric system, 10-13 
Mensuration. (See Measurements. ) 


Mixed numbers, 32, 55, 63, 66, 69, 73 
Multiples, 60 


Multiplication, of integers, 19-24 
Of fractions, 65-69 
Of decimals, 96 

Notation, integers, 3 
Decimal, 91 
Roman, tf 

Notes, promissory, 180, 16 
Discounted, 195-199 | 
Partial payments of, by U.S. 

rule, 182-187 
By merchants’ rule, 16 


PAGE 
Numeration, 2, 91 
Numbers, kinds of, L, 2,0 Os 
7, 20, 46 
Partnership, Pale iW 
Partial payments, 182-187, 6, 16 
Percentage, 149-215 
To find percentage, base, or rate 
per cent, 149-156 
Bonds, 201 ; Commission, 175-179 ; 
Duties, 188; Exchange, 205-209 ; 
Insurance, 173; Profit and Loss, 


154-159 ; Stocks, 199-202 ; 'l'axes, 


189; Trade Discount, 170-172; 
Rules, 4, 5. 
Powers, ad 
Present worth, 203, 6 


Principles of — 
Addition, -10, 59, 60; Cancellation, 
52; Commission, 175, 176; Dec- 
imal System, 2; Division, 27, 98; 
Greatest Common Divisor, 58; 
Interest, 160, 165, 169; Multipli- 
cation, 20, 21, 65, 97 ; Partial Pay- 
ments, 184; Profit and Loss, 155; 
Proportion, 214; Reduction of 
Fractions, 56; Right Triangle, 
222; Square Root, 218; Trade 
Discount, 171. 
Profit and loss, 
Promissory notes. 


155-159 
(See Notes. ) 


Proportion, 215-215, 6 
Pyramids, 225, 226 
Quadrilaterals, 116 
Ratio, 47, 77, 78 
Rectangles, Lae 


Review Exercises in — 


Fundamental rules, 17-19, 29-34, 


38-45, 50-53 
Fractions, . 62, 75, 78-90 
Decimals, 100-106 


INDEX. vr 


PAGE 

Measurements, 118-123, 181, 159- 
142, 232, 233 

Percentage, 159, 172, 179, 211 
Interest and bank discount, 167, 168, 
212 

Miscellaneous, 1438-146, 210, 237-262 
Rhomboids, 123 
Right triangles, 222 
Roots, 217 
Rules, 1-7 
Rule of Three, 33, 2138-215 
Savings-bank deposits, 192 
Signs, 7, 35, 47 
Similar surfaces, 229 
Similar Solids, 231 
Spheres, 221, 228 
Square root, 217-228, 7 
Stocks, 199-202 
Statement of problems, 24, 67 


PAGE 

Subtraction of integers, 12-16 
Of fractions, 59 
Of decimals, 95, 96 
Successive discounts, 171 
Surveyors’ measure, 9 
Tables for practice, 12, 14, 16, 29, 70, 
74, 154 

Tables of weights and measures, 8, 9 
Metric system, 10, 12 
Taxes, 189 
Time between dates, 165 
Trade discount, 170 
Trapeziums, 127 
Trapezoids, 124, 125 
Triangles, 126, 222 
True discount, 203 
United States money, 5, etc. 
Wood measures, 134 
Weights and measures, 8, 9, 10-12 


THE 
ESSENTIALS OF ARITHMETIC. 


Book ook IL. 


1.— Use of Numbers. What need of numbers has: 1. A mer- 
chant? 2 A carpenter? 3 <A farmer? 
4, A tailor? 65 A shipmaster? 6. A surveyor ? 


7. Speaking generally, for what are numbers used ? 


2. — Their Names. 1. Explain the meaning of their names 

from thirteen to nineteen. 2 From thirty 

to ninety. 8, The syllable -teen means what? What does -ty mean ? 
4, What does twenty mean ? 


5. What do we call ten tens? 6. Ten hundreds? 7 A thou- 
sand thousands ? 


8. Mention thirty numbers, each named by a single word. 
9, Show how other numbers are named. 


10. How many of these thirty words are used in telling the num- 
ber of feet in a mile? 


3.— Numbers 1, When we write numbers by the Arabic 
Expressed in Figures, system, how many different figures are used ? 
2. What people first used these figures ? 

3. What did the Romans use instead ? 
4, How is it that so many different numbers can be written with 


only ten figures ? 
1 


2 NUMBERS IN REVIEW. 


4.— A Decimal 1, In 5, 50, 500, 5000, how does the 5 
System. change in value? 2 What value has the 
zero? Why is it used? 3, The value of a 

figure depends on what two things ? 


4, In 505,050 name the orders of units. 5, Compare the value 
of each 5 with the value of the one next it. 


6. In place of the wavy line supply 
one or more words; in place of an 2, y, 
or z supply a number : 

100 ones=1 .; 50 tens=a hundreds; 
x hundreds = 10 thousands; 10..=1 
million; 10..=a hundred thousand. 


In a decimal system 
-ten units of any order 
make one unit of the 
next higher order. 


7. The money of the United States and of Canada has a decimal 
system of values. Explain what this means, by referring to dollars, 
cents, and dimes. Remember that decem is Latin for “ten.” 


5,— Reading Num- 407,826,903,531. 
bers. 1, Why are the preceding figures grouped 
in threes? 2. Name each period beginning 
with the lowest. 3, Read each period so as to show its value. 
4, Read the whole number. 5, What cider of units does the 9 
represent? Theo? They? Thed? 
6. Explain the use of the ciphers. 7 Compare the 3’s in value. 
8, Mention something counted by millions. 9, Can you think 
of any use for billions, trillions, quadrillions, or larger numbers ? 


6.— 1st: Oral Read- Read without using the word and: 
ing. 2d: Writingfrom 1, 4,705 6,157,008 42,200,020 
Dictation. 2. 27,003 3,000,975 34,000,739 
8, 195,006 600,001 93,040,760 

4, 70,590 17,080,005 349,000,672,084 


5. 104,509 3,864,219,461 16,080,372,094 


FIRST PRINCIPLES. 8 


7.— Writing Num- 1. Write the largest possible number, using 
bers. these six figures only: 0, 0, 2, 7, 3, 1. 
At sight or from Write in figures, putting a comma after each 
dictation. period before filling another : — 


2. 101 thousand. 6. Two hundred million, seventy-six. 
3, 1 thousand 1 hundred 1. 7. Ten billion, two million, sixty. 
4, 3 billion 108 thousand 11. _ 8, One less than a billion. 
5. 828 million 7 thousand 9. 9, Ten thousand hundreds. 
10. The sum of — 18000, 200000, 520, 6. 


8.— Test Questions. 1. What part of 4000 remains when we drop 
the last cipher? 2. When we annex a cipher 
we add _., and make the 4000 a times larger. 


38. Read 708 without the zero. How is the value of each figure 
changed ? 4, Putting the cipher after the 8 would change its value 
how ? 


5. What is the effect of moving a figure to the left? 6. To the 
right ? 

7. How would you increase the value of 478 a hundred times ? 
8, A thousandth part of 375000 is what ? 


9, Each cipher annexed to a number changes its value how? 
10, Each cipher removed from the right ? 


9. — Fractions. 1, What do integers contain, — whole ones 
or parts and fragments of whole ones? What 
kind of units have they? 2. What is a unit? 


3. Divide an object —an apple, a circle, a stick —into two equal 
parts. Is each part a unit? Why? 4 Is each part a whole unit? 
A whole 1? A whole half? 


5. How many fractional units are made by cutting into thirds ? 
Into fourths? Which are larger ? 


4 NUMBERS IN REVIEW. 


6. Units of any size smaller than 1 are called ... 7% One or 
more fractional units make ... 8, Give the largest possible frac- 
tional unit, and explain. 9, Give avery small one. How many of 
these make 1 ? 


10.— The Terms of 1, Can you illustrate in the divisions of 
a Fraction. the window sash, or otherwise, these or any 
other fractions ?—4, 7, 4, 4, 4, do dp + 
2. Read them in the order of their size. 8, How many of each 
would make 1? 4, Upon what does their size depend ? 
5. How many fractional units in 2, 74, 14? 6. How many of 
each size make 1? 7 Which fraction is nearest to 1 in value ? 
8, A fraction is expressed in what two terms? 9. Which shows 
how many units the fraction contains? 10. Which names them 
according to their size ? 


11, Give numerator and denominator and the use of each: —2; 


$3 Bios te yrs 7 wk.; 5); 73; 83% oz. 

12. An integer with a fraction added is called a ~~ number. 
138. How many units in $5,545? Which are integral, which 
fractional ? 


11. — Decimal _ 1, In the decimal system how may the 
Fractions. figure 1 have a certain value and then a tenth 
of that value ? 
Supply the omissions below : — 
sD. 10.0) eo ee 6 
S25 10 = ote ThO1 08 ten ees 
4, 01 =, of .= 7 8, 0.100 = . thousandths 
5. 0.01 = 54 of ..=7h Or .O 1 Ui temeaes, 
10. What denominators must fractions have that they may be 


written decimally ? 11. If not written, how is the denominator of 
a decimal known? 12. The decimal point is used for what ? 


1000L as cee ere 


FIRST PRINCIPLES. 5 


12. — Reading I. Read the following. Il. Give numerator 
Decimals. and denominator. Il. Give the value of each 
Jigure separately, 
1. 335; 3.6 4. 5757; 0.0038 T. 89,3 0.249 
2. z%53 0.54 5. 2543; 4.053 8. 0.1478; 0.090 
3. 253,; 2.09 6. 0.219; 0.765 9. 16.47; 18.476 


10. Compare number of decimal places with number of ciphers in 
denominator. 11. What are mixed decimals? Where is and used 
in reading them ? 


13.— Abstract and 1. Compare 7 and 5 with 7 days and 5 
Concrete Numbers. days. Which are easier to add? 2. Calling 
a number abstract when used by itself, and 

concrete when associated with something, describe the numbers in 
section 10. 

8. Classify: $275; 362 lb.; 873; one thousand; ten feet; a 
million people; a bushel and a half. 

4. What is it that you really multiply and add,—figures in ink 
that represent numbers, or numbers themselves? 5. In oral work 
do you add names of numbers or numbers themselves ? 


14.— United States At sight.—1. Why may 1.23 be read as 
Money. dollars, dimes, and cents, but not as yards, 
feet, and inches? 2. What is meant by a 


decimal system? (§ 4.) 
3. How many dimes are represented in $12.625? 4 How are 
they usually read? 5. How else may the 5 mills be read ? 


6. Read, and explain the use of ciphers: $7.77; $7.07; $7.7; 
$ 7.70. 
& 
Explain the meanings of dime, cent, and mill, as shown below : — 
7. Decem means ..; dime =a tenth of .. 
8. Century means a hundred; cent = a hundredth of 


9. Million means a thousand; mill = a thousandth of W. 


6 NUMBERS IN REVIEW. 


From dictation. —10. What is a double eagle? A quarter eagle ? 
11. Name four silver coins. 12. What is a nickel? 138. What 
other metal is coined? 14. What is a mint? 15. What is bullion? 
16. Why are not mills coined? 17. Of what use are they ? 


18. What is counterfeit money? 19. What gives value to paper 
money ? 20. Are U.S. coins made of pure silver and of pure gold ? 
21. Why is an alloy used? 22. Try to find what 18-carat gold is. 


15.— To be Read; I. Read as dollars, cents, and mills. II. As 
then Written from dollars and cents. III. As dollars and thou- 
Dictation. sandths. 
1. $38.19 4. $0.625 7. $6428, 
2. $5.194 5. $ 309.083 8. $2908 12, 
8. $ 24.074 6. $ 400.040 9. $80,076.95 


10. $1,014,806.09 - 
16.— Like Numbers. Mention one number of each kind :—1. Inte- 
gers. 2. Common fractions. 3. Decimals. 
4. Mixed numbers. 6. Abstract numbers. 6. Concrete numbers. 
7. Concrete integers. §8. Abstract fractions. 9. Concrete mixed 
decimals. 
10. Tell what kind of number, and the unit of each : — 
4ft.; $2; 2yd.; 2; $0.10; 18; 0.36; 16 ft. 
11. Select two or more with-like units; that is, of the same size and 
kind. 12. Define like numbers. 
13. Mention other numbers having the same units as those in 10. 
14. Give the integral unit and the fractional unit of — 
4i dozen 214 94 quires 2.1 seconds 
16. Which of the following numbers have the same integral unit ? — 
+ Ib. 4 oz. + ton 2,000 Ib. 1 ton 4 ewt. 
16. Change the unit without changing the value : — 
36 in. 6 ft. 12 ft. 120 sec. th. 4 wk. 


FIRST PRINCIPLES. 7 


17. DEFINITIONS AND SIGNS. 


[FOR REFERENCE. ] 


To the Teacher. — Having learned to understand and use technical terms, the student should 
be led to formulate his own definitions of them. Such as are given here may aid in securing exact- 


ness and brevity. 


Arabic System of Notation. So 
called because it came into Europe 
from Arabia, and was brought by 
Arabs from India. 

Decimal System of Numbers. 
A system in which ten units of any 
order make one unit of the next higher 
order. 

Decimal Fraction. One or more 
tenths, hundredths, thousandths, etc., 
of an integral unit. 

Decimals. Decimal fractions writ- 
ten after the decimal point, without a 
denominator. 

Decimal Point. A period used 
after ones and before tenths. 

Digits. The numbers for which 
the nine Arabic figures stand. 

Denominator. The lower term of 
a fraction. It names the fractional 
units according to their size and shows 
into how many equal parts the integral 
unit is divided. 

Fraction. One or more of the 
equal parts of an integral unit. 

Integer. A whole number of which 
the lowest unit is one, not any part of 
one. 

Mixed Number. An integer and 
a fraction taken together. 

Notation. The writing of num- 
bers in figures or letters. 


Number. ‘That which answers the 
question ‘*‘ How many ?’’ ; one or more 
units. 

Numeration. The reading of num- 
bers expressed in figures. 

Numerator. The upper term of a 
fraction. It numbers the fractional 
units contained. 

Period. One of the groups, of 
three figures each, counting from the 
units’ place. — 

Roman System of Notation. So 
called because invented and used by 
the Romans. 

Terms of a Fraction. 
numbers used to express it. 

Unit. One; a single thing. 

+ Plus; and; the sign of addition. 

— Minus; less; the sign of subtrac- 
tion. 

x Times; multiplied by; the sign 
of multiplication. 

~ or: Divided by ; signs of division. 

) In; a sign of division. 

—., /, (as in 1¥, 7/3) Signs of 
division. 

= Equals or equal; the sign of 
equality. 

$ Dollar or dollars. 

% Hundredths ; per cent. 

Ct., c., or ¢ Cent or cents. 

(@ At (the rate of). 

.*. Therefore. 


The two 


NUMBERS IN REVIEW. 


18. TABLES oF MEASURES. 


[FOR REFERENCE.] 


Counting. 


12 things = 1 dozen (doz. ) 

12 dozen = 1 gross (gro. ) 

12 gross = 1 great gross (g. gr.) 
20 things = 1 score 


24 sheets (paper) = 1 quire 
20 quires or \ Be 
480 sheets 4 me 


Time. 


60 seconds (sec.) |= 1 minute (min.) 


60 minutes = 1 hour (h.) 
24 hours =-lday (d.) 

7 days = 1 week (wk.) 
2 weeks = 1 fortnight 


30 (81, 28, 29) days = 1 month (mo. ) 


3 months or \ cee 
13 weeks = 
12 months or \ __ 1 year (yr.) 
365 days ~ (common) 
365d. 5h. 48 min.) _ 1 true or solar 
49.7 sec. } a year” 
366 days = 1 leap year 
10 years = 1 decade 
100 years = 1-century (C.) 
Value. 
U. S. Money. 
10 mills (mi.) = 1 ct. (ct., c., or ) 
10 cents = dime (di) 
100 cents or 
ee \ = 1 dollar ($) 
10 dollars = 1 eagle 


Canadian Money. 
100 cents = 1 dollar = $1 
English Money. 

12 pence (d.) = 1 shilling (s.) =$0.248 4+ 
20 shillings =1 pound (£) =$4.8665 
French Money. 

100 centimes = 1 franc (fr.) = $0.198 
German Money. 

100 pfennigs = 1 mark (M.) = $0.238 
Capacity. 

Liquid Measures. 

4 gills (gi.) = 1 pint (pt.) 


2 pints = 1 quart (qt. ) 
4 quarts =1 gallon (gal.) 
1 gallon = 281 cu. in. 


Dry Measures. 
(For grain, fruit, etc.) 


2 pints = 1 quart 

8 quarts = 1 peck (pk.) 
4 pecks = 1 bushel (bu. ) 
tres \ <1 barrel (bbl.) 
21 bushels j : 
1 bushel = 2150.42 cu. in. 


Weight. 
Avoirdupois Weight. 


16 ounces (0z.) = 1 pound (lb.) 


' 1 hundred- 
Hye ai bay (cwt.) 
2000 pounds or _..f iton*CT) 
20 Sea REN = { (short) 
2240 pounds = 1 long ton 


MEASURES. 


wheat or 


*G —_— 
60 pounds = 1 bushel { etiies 


*56 oie — fo Oe corn or rye 
*32 i 0 a oats 
196 * = barrel flour 


ot UR RNS 1 beef or pork 


* In most States. 


Troy Weight. 


(For precious metals, jewels, ete.) 


24 erains { mat yee) ny, 
20 pennyweights = 1 ounce 
12 ounces = 1 pound 
4373 grains = 1 ounce 
7000 ee = pound 
480 ‘ey Lounce 
5760 = = 1 pound he 


Apothecaries’ Weight. 
20 grains =1scruple (sc. or )) 
3 scruples = 1 dram (dr. or 3) 
8drams =1 ounce (oz. or 3) 


12 ounces 
5760 grains \ = 1 pound (lb. or tb.) 


Length. 
12 inches (in.) = 1 foot (ft.) 
3 feet = 1 yard (yd.) 
16} feet or 
Spars \ = 1 rod (rd.) 
320 rods 
5280 feet = 1 mile (m.) 
63,360 inches 


4 inches = 1 hand 


6 feet = 1 fathom 
6086.7 feet or 1 knot 
1.15+ con} . nautical mile 

mon miles 1 geographic mile 
3 knots = 1 league 


Circular Measure. 
60 seconds (!’) = 1 minute (') 
60 minutes = 1 degree (°) 

360 degrees = | circumference 
691 miles or 1° of latitude ; or 
60 geographic \- 1° of longitude 

miles on the equator 


Surface or Square. 


144 square inches \ = ie: square foot 
(sq. in.) (sq. ft.) 


9 square feet = 1 square yard 
(sq. yd.) 
501 square yards es 1 square rod 
2721 square feet (sq. rd.) 
160 square rods . 
43,560 square feet f sore 
fal ated am { 1 square mile 
(sq. m.) 
1 mile square = | section 
36 square miles = 1 township 
1 square 
100 square feet — (in roofs, 
floors, etc.) 


Solid or Cubic. 
1728 cubic inches \ i { 1 cubic foot 


(cu. in.) (cu. ft.) 
27 cubic feet \ _ lcubic yard 
(cu. yd.) 


Wood Measures. 
16 cubic feet = 1 cord foot (cd. ft.) 
128 cubic feet 


8 cord feet Bed COrdst ec.) 


10 NUMBERS IN REVIEW. 


Addition. 
19. — Combining 1, Give the sum of 7 and 8; then prove by 
Like Numbers into counting. 2 If you had not learned the sum 
One Sum. of 8 bu. and 6 bu., how could you find it? 
At sight. 3, Find and explain a quick way of adding 


6, 41, 3, 1,59, °2, 5. Oe DIC women 
think more important: to add rapidly or accurately ? 


Explain what change you make before adding — 


5. 3wk.andi4da 6, 3 yd., 7 ft., and 
v41n, l; »13 ands 0.Sb: 

8. In number 6 why not change to 
inches? 9, In number 7 change to cents 
and add. 

10. 2 gal. +6pt.=aqt. 11. 48 oz.+ 
Zep 1D) MAA My tet OO. ie ce ey 
13. 4+3=7%. 14 9 ft.+108 in? 
15. 6 yd. + 36 ft.? 16. 1 h.+300 min. ? 
17. Why not add 4and+? What change must be made? 


Before unlike num- 
bers can be combined 
into one sum their 


units must be made 
alike. 


20. — Practice in Give in thirty seconds 48 sums of two digits: 
Rapid Adding. 1, By columns of two each. 
At sight. 2. In pairs along the line. 


3. Add the four digits in each square : — 


20 Vl Zee 


Me 13 | $2146 16 1520 15.07 226 OF omiee 
1 5)}1 6)4 6)6°9)2 8) 2 718 84 697-19 57S O7 br oheeD 
B. |74)/75)4 9/4 2)4 4/5 8)4 2/4 5/38 3}1 6)1 7/2 4 
7619 618-911 719 518 91S) 4p io (oem 


4s Add-by 10's) 8: O12 stay 3 Ose Geis mero ae 


21.— Rapid Adding. 


<2 200 SS SS ae ee 


At sight. 
46, 34 10, 
19, 71 inh 
53, 47 12, 
86, 32 13, 
38, 69 14, 
47, 46 15, 
65, 25 16, 
32, 99 17. 
12, 88 18, 


22. — Written 


Addition: the 


Process Described. 


421 
635 


442 
340 


390971 


set down at the left. 
Why are they kept separate ? 


value. 


23.— Written Ex- 


EXERCISES. 


Practise 


instantly. 


127, 123 
900, 140 
560, 240 
767, 232 
808, 191 
346, 509 
BRS O12 
694, 106 
333, 766 


until 


you 


19, 
20. 
21. 
22, 
23, 
24, 
25. 
26. 
27, 


Can 


give 


11 


“ 


each sum 


3000, 1798, 2000 
1300, 2000, 175 
4080, 1507, 6000 
85, 800, 9000 

Ts) Te Ts Ts 

0.06, 0.18, 0.24 
25%, 8%, 30% 
0.41, 0.19, 0.40 
$4.75, $3.25, $7.87 


Give directions for five steps in adding 378, 


V. Testing. 


492, 864, 793, 956, 309. 
I. Arranging the numbers. 
II. Beginning to add. 
III. Setting down the sum. 
Visa Garry ine. 
The separate sums of four long columns are 
Read each to show its 


378 
492 
864 
793 
956 
309 


3792 


Without copying, first add vertically; then 


ercise. horizontally. 
F 2. 3, 

. $3.47 $ 14.69 $ 193.67 
8.62 48.96 846.84 
9.46 o7.81 932.71 
6.58 47.94 683.77 
7.39 82.66 865.75 
9.88 68.45 392.50 


$ 


4, 
4769.83 
4592.16 
8437.66 
6989.84 
4329.41 


6832.47 


5, 
$ 6483.47 
8432.97 
6432.98 
8469.32 
9396.48 
9375.58 


Ee NUMBERS IN REVIEW. 


24.— Rapid Adding. Try by practice 


Written. to do one example A. B. 
a minute. Without copying add Ist, in A; c. $475.21 | $648.90 
2d, in B: — . 649.85 938.27 
> 837.64 642.85 
1, From c¢ to / inclusive. - 946.89 937.63 
2ptoy 1% dtog 14 ftov . 987.63 | 846.75 
3. dtom &, ctor 13. e tow Airo 21.14 324.93 
4, eton OF 182088 14, dtoa » 608.55 698.79 
. 327.83 128.93 
5, ig to o 10,7 tOrG 15. c to y » 469.75 648.72 
6. etonp, V ll giow . 984.96 562.37 


678.94 689.85 


627.34 283.97 


Subtraction. 234.56 135.42 


25.— Taking a 1, One part of | ». 789.12 | 698.57 
Part away. 17 eggs is 8 eggs. . 846.89 | 569.38 

How would you | » 764.83 | 783.92 

find the other part if you had forgotten 758.75 964.83 
that 8 from 17 leaves 9? 839.65 385.75 
2. Take out 10 stormy days in Janu- . 087.93 978.59 
ary: x remain. 931 less 10, or — 10, or = OOABA 628.32 
diminished by 10 = a. . 849.64 759.67 
38. Make a problem in subtraction, 376.86 314.11 


using concrete numbers. Which is sub- | ¥% 978.35 629.55 
trahend, which minuend? 4, The other 
partis .. The largest part is 


5. ‘In subtraction, which terms must be like numbers ? 


26.— Finding the Subtract and say whether the result is re- 
Difference. mainder or difference : — 


1, You have $12 and spend $7. 2. You have 
$12, and I have $7. 3. You have $9, and will earn enough to make 
it $ 15. 


EXERCISES. 13 


4, How do you find the third term when you have the difference 
and the subtrahend? 5, The minuend and the difference ? 


27.—The Terms 1, Which is the larger number, 3 ft. or 

in Subtraction must 24 in.? 2 Which is the larger quantity ? 

be Like Numbers. 3. How can one be subtracted from the 
At sight. other ? 


4, A boulder weighs 7000 1b., a stone 
block 4aton. The difference in weight is x Explain the process. 


Where you can, give two values to x, first like the minuend, then like 
the subtrahend : — 


5. 4 lb. — 32 0z.=2 9, 10 h. — 240 min. = @ 
6. 60 mo. —2yr.=2 10. {-—i=2 
7. 0.7 —0.03 = a ll, 2T.—21b.=2 
8. $250 + a= $525 - 12. «— $166 = $ 34 
28. — Rapid Give in one minute or less the difference 
Subtraction. between each number and the one below it; 
Oral. _ between each number and the one at the 
right of it. 
A, | 11 9 | 13 7 | 10 Looe ely oo) 18, 1d 
| 2 ‘Ti eed ad 8 | 6 bao 7 On ae 5 
B. 9 O05 216.) 11 ae ar: Nata a Bees Cs Fe eb 8 16 
3 iro S46 341° 9 (AGS S44 9 
0; | ib ices OAM ey ela Lb | foe eile tte 8 119. °° 10 
7 S14 5| 6 9| 8 7 Pep Bei. 7 
Pema 0 1AT 18d 1316 918118 12) 8~ 12 
a Et Rane: al i 


| 
| 
} 
/ 
| 


14 NUMBERS IN REVIEW. 


29. — Rapid - |. Give the difference between 100 and 
Subtraction. each of the following numbers. 2 Between 


Sight or dictation. each number and the one at its right. 8. Be- 
tween each number and the one below it. 


a) 11 (88) 44 74.152. 709 SO" DS 60) OF eal es ela 
b| 35. 61.) 82 14) 91> 33 | 22°65) 42 163410, 59 Oo eae 
ce | 88 30] 23 57 | 89 48] 95 29) 68 32) 84 26) 79 16 
O.) 05 81) T3° F221 94 O65 OOS. 03 ee OO neon me 
é. | 92 19.1 63: 45>) 96. “1857 86) 46097 Oe 49 eT Ola es 
. | 64 77 | 24 28) 34 39] 78 38) 80 25) 97 54] 98 40 


4, From 1000 take 120 175 225 350 760 807 901 


5, What remains when each of the numbers in the table is taken 
Gut Olvl29 7 dole li3r 


30. — At Sight. 


1, 2, 3, 4, 5, 6. 
700 3000 60503 25000 oo111 36459 
325 800 40402 371892 46221 47560 


7. 34+ 2=48 x + 27 = 80 « — 79 = 23 x + 24 = 150. 
8. Replace x with the proper number. Think quickly. 
Minuend 48. 62) ~@ (840° 27e a 80. abe = eeu 


Subtrahend 16 x LOR PALO x 46 x Lio 4 dee 
Remainder x A aes x ab! Bee aks 19 x Lied i 


—— 


31.—OralProblems. 1. If you sleep 8 h. and spend 54 at school, 
how many of the 24 remain ? 

2. Out of $2 I spend 37¥, a half-dollar, 
and adime. What have I left ? 

3. What is the hundredth day in 1897 ? 


4, What is the difference in latitude between a city 35° north 
of the equator and one 34° south of it? 


For dictation. 


QUESTIONS AND PROBLEMS. 15 


5. If a person is 69 yr. old to-day; when was he born? 6, A man 
who died in 1879 would have been 100 yr. old if he had lived 18 yr. 
longer. When was he born? 7 In what year was a house built 
that now lacks 12 yr. of being 150 yr. old? 

8, What number is 16 less than 100 — 59 ? 

9, At 72 min. after half-past three, what time is it ? 

10. Find what remains in counting backward by 13’s from 100. 


32. — Oral 1, From 97 count backward rapidly by 
Exercise. Gea e- os. bas, 
At sight. 2. Count up to 200 by 17’s. 


3 1384+2=120 250—x=120 2424=17 
What change from a $5.00 bill that pays for — 


4, Oysters, $0.75 5. Gloves, $1.25 6. Pens, $0.35 
Crackers, 0.38 Scarf, 0.75 ipke Mth 
Cheese, 0.62 Pin, 2.50 Paper, 0.874 


7. Add the difference between 38 and 67 to the subtrahend. 


Find what remains after receiving and paying as shown below : — 


Received. Paid. Received. Paid. Received. _—_ Paid. 
8. $1.16 $0.93 9, $45.00 $28.00 10. $2.25 $1.75 
0.24 O17 95.00 19.00 3.75 2.30 
0.60 0.25 70.00 23.00 De201 bac 
33. — Written 1, Try subtracting one order at a time, in 
Subtraction: the work at the left, giving each figure its 
the Process. real value. What is the first difficulty ? 


457 from 683. 2. If you had 83 sticks in bundles of 10 each, 
Minuend 683 With 3 sticks over, how would you subtract 7 
Subtrahend 457 sticks? How many bundles remain ? 

38. At the left, 5 tens are to be taken from 2 
tens. 4 What was added to the 3? 65 What 
may be added to test the work ? 

6. Give directions for each separate step in the process. 


Remainder 226 


16 NUMBERS IN REVIEW. 


34. — Written Without copying find quickly the sum of the 


Exercises, four differences between — 
1. eand f 3 gandh 6 tandj 7. k and J 9, manda 
%. fand g 4, h andi 6. 7 and k 8, 7 and m 10. » and e 
Find the sum of the ten differences between — 
11s Arand?B 13. Cand D 15. A and C 
12, Band C 14, Dand A 16. Band D 
A, is) C. D. 
e. $3764.82 $ 4769.31 $ 5000.37 $ 9000.15 
ap 927.35 3468.97 689.82 794.38 
q. 860.83 385.68 1348.75 1866.75 
h. 1527.96 2487.32 946. 2889.43 
1. 3184,98 694.39 37.89 648.95 
j. 2876.45 1748.64. 9586.54 1864.37 
k. 825.35 4839.87 829.85 624.94 
l. 96.47 658.34 1472.98 1739.41 
m. 849.383 1987.62 468.52 866. 
nm. 3276.41 594.83 5500.31 49.75 
35. — Oral 1, Four parts of 75 are 18, 9, 13, and 22, 
Problems. The fifth part is a 2. 387 gallons are in a 
At sight. tank. Add 17 while 23 run out. What re- 


mains ? 


8, An engine goes forward 25 rd., back 88 rd.; forward 60 rd. 
How far is it from the starting-point ? 


4. How much farther is it round a 17-foot square than round a 
square 13 ft. wide ? 


5. By annexing to 57 the figure 6 how much is added ? 
6. Taking the 5 from 275 leaves how much ? 


7. Bought a pony and phaeton for $500. Sold the pony for $175, 
losing $50, What did the phaeton cost ? 


QUESTIONS AND PROBLEMS. 17 


8. Having $400 in bank a person draws $25, deposits $150, 
draws $75 and $50. How much remains ? 


9, One horse is worth $50 more than a second and $150 more 
than a third. If the highest priced one is worth $200, what are 
they all worth ? 


10. If you find 5 eggs one day and 6 the next, how many dozens 
will you get at that rate in a week ? 


36. — Problems. 1, How much remained in bank to Mr. 
Rich’s credit Saturday night, what he put in 
and took out being as follows for the week : — 


Deposits: $26.95, $793.82, $427.96, $ 839.64, $500, $ 387.28. 
Withdrawals: $18.56, $ 689.37, $ 419.28, $ 649.39, $ 600, $125.82. 


2. A merchant’s assets are as follows: — 

Merchandise in store, $ 24876.39; cash on deposit, $ 1489.38 ; due 
from customers, $4897.64; real estate, $28649.27. He owes for 
merchandise, $ 16483.56; for real estate, $6498.27; on promissory 
notes, $ 6483.75. How much will his estate be worth if he closes out 
his business and pays his debts ? 


Written work. 


3. I have on hand at the opening of business cash to the amount 
of $846.95. I pay out $84.92, $64.87, and have on hand at night 
$ 837.69. What have I received ? 

4, I received during the day $ 249.85, and I paid out $521.75. 1 
had on hand at night $37.62; in the morning «. 

5, Thomas Bond begins business January 1 with cash $ 478.37 and 
merchandise $1875.28. At the close of the year he has $1487.63 
worth of merchandise and $738.29 in cash. How much has he 
gained, or lost, during the year ? 

6. The sum of two numbers is 346301. The smaller is 89795, the 
larger a. 

7. What number must be subtracted from one million to leave the 
difference between 347689 and 486931 ? 


18 NUMBERS IN REVIEW. 


8. The distance from A to B is 628 feet, from A to C 1426 feet, 
and from B to D 1648 feet, all in a straight line. How far is it from 
C to D? Draw a line and mark off the distances. 

9. How many days of 1897 have passed before Aug. 15 ? 


10. What is the difference between the sum of column A, page 12, 
and that of column B? 


37.— Cash Nortr. — An account with ‘‘Cash”’ is, as it were, an account 
with one’s cash-box or pocket-book. Cash is debtor for all that 
Accounts. is put in, and cash is credited with all that is taken out. 
Dr. CASH. Or. 
1896 1896 
Nay /\@n hand /00\00 Nay o by Ndaé. bought L50\00 
O|\So Rent ree’d|| 50\00 Ale Kano “ \|250\00 
7\ “ Mdae. wold\| 25\00 ea) Clothing “7 25|00 
VO CO ™ Lane glivoed //\ batanee || 75|\00 
ie GOO\00 GOO|00 
Nay 22 bn hand 5|00 


1. Cash is charged with having received four amounts which it 
owes me and for which it is my debtor. How much was there at 
the beginning ? 

2. What was added from sales of merchandise? 3. Cash is debtor 
for the price received for land. Why is the income from rents charged 
to Cash? 4 What is the total amount my cash has received if I - 
wish to draw upon it? 5. How much does Cash pay back to me 
for the piano purchase ? 


6. Why do I credit Cash with my clothing expenses? 7. What 
are the total outgoes for the month ? 


8. What is the footing of the debit side? 9. What more might I 
have spent so as to balance the footings? 10. How is the balance 
found ? 


QUESTIONS AND PROBLEMS. 19 


38. — Written 1. Balance the cash account of Charles 
Exercise. Watson. He has on hand $4.21. He re- 
ceives at various times $6.24, $7.36, $8.49, 

$7.34, $6.75. He pays out $8.75, $9.81, $3.26, $8.39. 
2. Monday morning a merchant begins business with $247.84 on 
hand. He receives $24.75, $86.91, $ 84.28, $97.25, $164.29. He 
pays out $18.99, $ 37.49, $64.91, $83.15. Find the balance on hand. 


Find the balance of each of the following accounts : — 


3. 4. 5. 
Dr. Cr. Dy. Or. Dr. Cr. 
$ 987.65 $629.55 $4768.82 $468.34 $649.81 $82.46 
1839.76 83.74 947.61 984.59 8439.87 981.32 
6482.91 968.71 847.77 1483.22 648.38 641.25 
478.85 28.46 3998.64 91.76". 239.86 
698.47 318.93 8372.91 728.41 


Multiplication. 


39. — Numbers A: Unequal numbers. 8B: Equal numbers. 
Combined. 94+847+4=28. 7474747, or4x7, =28. 
At sight. Il. In A, the combining process is \.. 


2. Can more than two numbers be added at a time? 

3. Under B, the first process is ..; the shortened process is W. 
4. Do you know the product of 4 7’s by counting or from memory ? 
5. Which number is to be multiphed ? Which is the multiplier ? 

6. Why not get the result in A by multiplying? 7 Compare 
addition and multiplication. 


40.— Rapid Factoring. What numbers multipled together, Le. 


Oral. what factors (smaller than 14), produce — 
1. 28, 32, 33, 35 4. 65, 66, 72, 77 TeUB ALLO MIT 
2. 36, 39, 42, 45 5. 78, 81, 84, 88 8. 121, 130, 132 


8. 48, 49, 52, 54 6. 91, 96, 99, 104 9. 143, 156, 169 


20 NUMBERS IN REVIEW. 


10. The sign means: separate into two equal factors, or find 
the square root. W25; V81; V36; V49. 11. Vi44=axy; 
V121 = 2; V64 = x; V4 x 25 = a. 


41. — Principles in xx $8 = § 24. 
Multiplying. ASX ee Were eis I. Only one factor 
At sight.—1. Say | can beconcrete. Both 

which is multiplier and which multiplicand, | may be abstract. 


giving values. 2. What are the factors Il. The product 

(makers) of $24? Of 8 sq. ft. ? and the concrete fac- 
For dictation. —8. What is anaddend? | tor will be ‘like’ 

A subtrahend ? A multiplicand? | numbers. 

4. Make two examples: the multiplicand Ill. The order in 

concrete in one, abstract in the other. | which the factors are 

5. Try multiplying by 5 stones or any | used will not affect 


concrete number. the product. 
6. Show with objects that 3x4 of a 
kind are 12 of the same kind. 7. Give the factors of $21; 49 m.; 
18 cases. 
8. Compare 4 x 5 bu. and 5 x 4 bu. 


42.— Rapid For dictation. —1. 9 and 12 are factors of 

Multiplying. what? 13 and 6? 19 and-3?. d1land 122 
Oral. $8 multiplied by 4? 41? 41° 

Give the product : — Multiply by 9 and add 9 :— 

8. 3,6%; 10%, 2; 12, 13. Glee S 14> G4GLD oie eles 

4913 X POND x Pate ol. Give two factors making — 

0. 374, 2; 5,124; $14, 4. tT. 63772748 Ol thy fe Sood, 

8. Multiply the following by 8; by 9; by 12: — 


7 bales 70 bales 80 rods 50 fathoms 90 feet 
At sight.—9. Give each product quickly, stating which factor 
is multiplicand : — 
$800 600 700 rd. 900 800 m. 400 6000 
) I 2ivd. 76s 9.30). Pe ula 16m. 13 


————sieg  ) oemenanmnaioniceeniiot 


QUESTIONS AND PRINCIPLES. 


10. Give two factors of 182 sec.; 125 in.; 144%; 108 h. 


IL; Take 4 x 7 from 9 x 7. 13. Add 18 x 13 and. 2'« 13. 
12. Take 6 x 8 from 8 x 9. 144, 3x9in.+6x9in=®@. 


15. 8'=3 x 3=9; 4?; 67; 88; 72; 9; 122; 207; 502 


21 


43.— 1st: Written Arrange as shown at the right, then complete 


Work; 2d: Oral Ex- the equations. 1. Find 8 x 858. 
planation. ei fe aiye & 112x543 = 8x 8= 
Think what is required and write equations from the oe 0 
following datu: 3 x 9¢ = 27¢. Pee 
4, At 2 for a quarter what will 14 baskets cost ? 8 x 858 = 


5. 387 poles at $4 each. 8. 1031 lb. of 6¢ sugar. 
6. 6 doz. barrels, $2.50 each. 9, 8.h. 20 min. a day, 6 days. 
7. 12men,10d.,$2aday. 10. $1 a week for 2 yr. 


44.— Written By an integer of one figure. — 

Multiplication. 1. Under C what are added to get 

For oral analysis. the result? 2, Explain their position 

and real value. 8. Show how the re- 

A. B. sult may be got without setting down 
183 183 the partial products. 


By 10, 100, 1000, ete. —4. Annex 
a cipher to 28, and give the values 
of the 8 and 2 before and after the 


2954 


change. 6. How would you multiply by 


Every cipher an- 10,000? +6, Compare the work under A 
nexed to an integer and B, and make a rule for multiplying by 


multiplies by 10. any number of 10’s, 100’s, ete. 


10, by 100, by 1000:— 


37 64 8357 946 3472 8476 29,475 


8, Compare 90 x 100 and 100 x 90. 9. 120 x 300 =. 


7. Read these numbers multiphed by 


rays NUMBERS IN REVIEW. 


45.—For Rapid Fig- Find the product :— 
uring. 1, 67x 4763 =a 2, SO eo Leesa 
and 6789. 4. 5987 by 7. 5, 84,965 by 80. 
6. 8 x $12,039. 7. 848,794 and 300. 8, 1,203,900 x 50. 9 1 ft. 
Sin. x 3100. 10. 34,000 and 90,000. 


46. — Multiplying 1. In the work at the right, read the multi- 


by any Integer. plier. 2 What three partial multipliers are 
For oral analysis. used? 38 Read the 2d 

partial product. 4 4 x 578 578 

would be what? 5, Explain how the 3d partial - 346 


product is got. 6. Would it affect the result if we 6 x%578= 3,468 
should multiply first by 300? 7% In ordinary 40x 578= 23,120 
work how much of this may be omitted? 8 Where 300 x 578 = 173,400 
is the lowest figure of a partial product tobewritten? 346 x 578 = 199,988 


47.— The Process Give directions for six steps in multiplying : 
ESE le 1, Arranging the factors. 
Choosing a Multiplier. ae : 
2. Beginning to multiply. 
Gee 3. Setting down and carrying. 
$7 multiplied by 4, Arranging partial products. 
2378 must equal 5. Finding entire product. 
16646 . 
% 6. Testing the work. 


For 2378 ; 
paulerplied ir 7 a. In finding the cost of 2378 bbl. flour, at 


equals 16646 7, what is the true multiplicand ? 
b. To shorten the process, why must we use 
abstract numbers as shown at the left ? 


48.—For Rapid Figuring. 3, 37¢ x 432 7. 64 oz. x 976 
4, 427 x 83 Geet oe 
1. 5386 x 846 5. 329 x 847 9. 846 x 9372 


2. 3976 x 597 6. $687 x 4395 10. 387 T. @ $8 


: 
| 


ANALYSIS. 23 


49. — Examples Oral.—1. Perform the work at $4.37 
with Decimals. the right aloud, giving each figure 19 
its value in dollars or cents. 2, Ex- $39.33 
plain the position of the decimal point in the product. $ 43.7 
$ 83.03 
Written. — 3. 479 ed. @ $12.00 ? Write the product in cents ; 
4, 4868 T. hay @ $27 ? then in dollars :— 
5, 25,789 bbl. flour @ $5.00 ? 9. 239 men get $5.75 each. 
6. 787 M. brick @ $16? 10. 2958 lb. tea @ §$ 0.67. 
1. 193 x T cents ? 11, 234 million half-dimes. 
8, $0.07 x 795? 12. $5.623 x 8 x 3. 
50. — Oral Explain the process that you use :— 
wi te ake 1. Take 16 in. from. yd. 2 Add 1 yd. 
and 4+ ft. To what may both be reduced ? 
8. 7x 4.0f 120 ='2. 
4, =, of $57 =a ond ae cate} bee 6. x= 200 x 84 
qin Of GST =y Ne OUU ane yY = 71, of 2375 
10 x $0.57 =z 13 x 800=-2 BLEU 
ie 10 gh a7 gal. 8. 2 yd. @ 18¢ 9, 31 lb. @ 20¥ 
LOS i=: “yd. 1 doz. @ 16¢ 3 lb. @ 16¢ 
4. Ib. = & 07, 1 yd. @ 32¢ 2 qt. @ 121¢ 
< lb. = @& 02. 


Te a O45 OV OF er, 


51.— Analysis of 1, Make a problem in which the equation 


Problems. 4x 162¢=~2 will indicate the work to be 
Oral. done. What are the two equal quantities ? 


2. Which is easier to do: reason about a problem so as to show 
how it may be solved; or figure out the result after being told how 
to do it? From which do you learn more? 38 Uxplain the maxim: 
“ Well understood is half done.” 4 Define an equation. 


24 NUMBERS IN REVIEW. 


od 


52.— Problems to 1, $2.50 was the expressage on 19 tables 
be Stated. (@ $12.74, and 28 chairs @ $2.58; a is the 
For written work. total cost. 
Statement. — $ 2.50 +19 x $12.74 + 28 x $2.58 = a. 
Make an equation showing all that must be done to find the value 
of x; then find tt. 


2. 130 men @ $2 a day, 47 @ $1, and 8 @ $ 3.50 receive x dol- 
lars in one day. If paid weekly, they receive y. 


3. w is what is paid for — 4, The amount received = a. 
24 and 64 lb. tea @ $1.19 217 A. wheat ; 
59 bu. potatoes (@ T5¢ 27 bu. to an acre; 
30 Ib. coffee @ 34¢ sold for 83¢ a bushel. 


5. A nursery contains 1000 trees; 75 are dead; the rest are to be 
sold @ $2 each. They will bring $a. 


6. 3 house lots cost $1260.80 each and sell for $1500 each. The 
total gain is a. 

Use a short method in finding — 

7. The sum of 827 x $9.28 and 183 x $ 9.28. 

8, The difference between 649 x $12.84 and 149 x $12.84. 


Division. 
53.— Finding an For dictation. —1. The product of two 
Unknown Factor. factors is 48. One is 6; the other, —. 
2. How many 9’s in 54? What goes 9 times 
in 63? 3 12 is the multiplicand; how many times is it taken to 
make the product 84? 4, What multiplicand, repeated 12 times, 
makes the product 108 ? 


5, Suppose one factor and the product are known; how is the 
other factor found ? Tlustrate, using $6 x a= $42; and $a x 5= 
$45. 6, Why is the process called Division? 


PRINCIPLES AND PROCESSES. 95 


At sight.—7. Show by the examples in 5 that — 

(a) The product becomes the dividend (something to be divided) ; 

(6) The known factor becomes the divisor ; 

(c) The unknown factor, when found, becomes the quotient (show- 
ing how many times, or the size of each part). 


8. Give the quotients: «x15=30d.; $9 x2x=108; 
= 
oO, 96+-24=a@; T2:2=4; alae eG 


18. Pe he 
9. Describe the four different ways of indicating division shown 
in the preceding line. 


10. Find two factors of 96 leagues; 91 d.; 168 h.; 182¢. 


54.— The Process 1, Division is the reverse of ... 2 How 

of Division. may the multiplication table help to find a 

) quotient? 38 Without that table how might 

one find the number of 12’s in 60? 4 Find by subtraction the 
number of 24’s in 96. 


5. How many 12’s in 1740 ? 


A. B. C. 
145 
12)1740 12)1740 12)1740 
1200 = 100 12’s 12 , 145 
pesca f 6. Are there 200 12’s 
sath e nei in 1740? 7. Are there 
60 60 100? 8, Subtract them: 
60= 65 12's 60 


what remains? 9, How 
many 12’s in 540? 

10. Subtract 40 12’s: what remains? 11. 60 =@ 12’s; subtract 
them: what remains ? 

12. How many 12’s in all have been taken from 1740 by the three 
subtractions? 18, Explain the changes under B. 14 Perform 
aloud the work of C. 


Total = 145 12’s 


26 NUMBERS IN REVIEW. 


15. What is the difference between long and short division ? 


A. be 16. Explain process 

207935 A of finding how 

235)48674 235) 48674 many 230’s in 48674. 
47000 = 200 235’s 470 | 17, Explain process B. 
1674 1674 18. Why are there no 
1645 = 7 2835's 1645 tens in the quotient ? 

29 207 235’s 29 19. What part of an- 


other 235 is found in the remainder, 29? 20. By how much 
should the dividend be increased to give 208 for a quotient ? 


55. — Examples. 1, How many 9’s in 4752? 2 8’s in 

Written. 9896? 38 12’s in 3300? 4 15’s in 4650? 

5, 25)16325. 6. 17784+312. 7% 19998 

8. 27 x «= 40527. 9, Product= 9672; quotient = 372; divisor =a. 

10. 96 and 75 are the factors of what dividend? Give proof. 

11. Show that 33810 + 245=138. 12. Multiply 245 by 138 and 
find the partial products in the work of Example 11. 


56.— For Dictation. 1. How many 99’s in 500? 1000? 10,000? 

Oral. 2, Count by 99’s to 500. By 98’s to 500. 

8. Define dividend; divisor; remainder. 

4, 4 of 12 ist of what? 5. $12+24=$2. 6, What kind of 

number is the quotient when the divisor exceeds the dividend? 
7. If 6 shillings make $1, one shilling is worth a cents. 

8. Knowing one factor of 48, how can you be sure of the other? 
9, What are the three factors of 18? Give 4 divisors of 18. 10. The 
largest divisor of 72 is a Of 48? The greatest divisor of both is 
what ? 


57. — The Given Fac- 1, How many $10 bills make $300? 
tor Like the Dividend. 2. One factor of 60 yd. is 6 yd.; the other 
At sight. is .. 8 In multiplying two factors to 
) make a product, which factor is always 

abstract? Which may be concrete ? 


PRINCIPLES AND PROCESSES. ve 


4, If 7 ft.x 6=42 ft., 42 ft. 


+7 ft. =a, and 1 of 42 ft. =y. A divisor that is like the 
5. Show by the last example dividend is one of its equal 
that when the dividend (or prod- | Parts. The quotient tells their 
uct) and the given factor are alike number, showing how many 
the factor to be found must be times the divisor can be sub- 
abstract. 6. Show that the quo- tracted from the dividend. 


tient might be found by repeated 
subtractions. 


7. 90 in. +6 in.=@. 
8. 28 qt. in 84 qt. # times. 10. 45% +15%=~2. 


58. — The Given Fac- 1, 8 hats cost $ 40, 1 costsa. x«x8= $40; 
tor Abstract; Dividend 1 of $40=a. 2 When a product or divi- 
Concrete. dend is concrete, are the factors like or 

For dictation. unlike ? 


3. Which factor shows — 

(a) the number of equal parts 
united ? 

(b) the fractional part of the many equal numbers make 
dividend to be found? the dividend. 


(c) the size of the parts? The quotient is one of 
these numbers. 


9, 400 ft.)8000 ft. (a 


An abstract divisor of a 
concrete dividend shows how 


4, Kach woman gets ;, of $60. 
The number of women is @. 
5. When 20 books cost $200, what part of it will one cost ? 


59. — Examples. 1. 6482 ft. + 90 ft. = m. 
Written. 2. One factor of $475,000 is $250, the 
other is 2. 


. 84 equal numbers make 6300 yd. Find one. 

360 m. = multiplicand; 25,520 m. = product; # = multiplier. 
Multiplier 125 ; product 100,000 bales ; multiplicand a. 

. Divisor x quotient = $16,750; divisor = $ 670, quotient = 2, 
. Dividend = 197 x $461; quotient = a, 


AO - © 


98 NUMBERS IN REVIEW. 


8. After subtracting 220 atimes from 44022, what is the least 
that must remain ? 


Find cost of one acre, when — 


9, 64 cost $ 367 x 28. 10. 128 cost $ 1000. 

60. — Division: the 
Process Described. ah ag fee Sit 
144 pens)10,000 pens 25)%176.95 

8640 pens $175 

1, A box of pens 1360 pens $1.95 
contains a gross. Make 1296 pens $1.75 
a problem for the work 64 pens $0.20 


under A. 2 Under B. 


38. Explain why in each case all the numbers but one are like 
numbers. 4 What numbers when added make the dividend ? 
5. Why not divide the $0.20? 


6. Give directions for six steps in division. 
J. Arranging the numbers. 
II. Choosing 1st partial dividend. 
Ill. Writing quotient figure. 
IV. Finding product to subtract. 
V. Completing 2d partial dividend. 
VI. Finishing the process. 


61. — Rapid First column (p. 29). 1. Divide quickly by - 
Division. 2; 3; 4; 5; 6. Give integral ea and 
Oral. remainder. 2 Find 4+; +; 4; 4; 7. Give 


the exact size of the equal parts. 

Second column. 8 In each number how many times will 10 go? 
100? 200? 300? 40? 4 Use as divisor 50; 60; 80; 90; 70. 
Give remainders. 

Third column. 6. Give quotient and remainder in cents after 
dividing by $0.50; $0.25; $0.80; $1.10; $1.20. 

Fourth column. 6. Give results in dollars and cents of each 
dividend + 1,000; + 2,000; + 200; + 4,000; + 3,000, 


EXERCISES. 29 


1. 2. 3. 4. 5. 6. 
a. 21 50 $1.50 $ 4261 $ 567.82 347,694 
b. 32 520 1.25 8937 739.75 932,976 
c. 43 636 1.75 6425 947.50 843,207 
d. 54 724 1.38 8034 842.90 600,898 
e. 65 837 2.75 6481 838.38 347,291 
ooo 964 3.25 8972 496.81 468,394 
qg. 87 523 4.50 4729 149.85 729,831 
h. 98 649 5.40 6834 328.74 476,984 
i. 89 732 9.60 9287 692.48 294,765 
a (8 807 7.23 3199 728.47 300,041 


62. — Practice in First column. 1, Divisors: 18, 14, 15, 16, 17. 
Division. Use short division. 
Written. 2. Divide by 18, 19, 20, 21, 22. - 

Second column. 3. By short division find the other factor when 
mmeris 00,93, 97, 96,-95. 

Third and Fourth columns. 4 Change both numbers to cents, 
then use the larger as dividend. 

Second and Sixth columns. 6, Divide numbers in 6 by numbers 
in 2 and give remainders. 

6. Col. 4+ (col. 3+ col. 5). 7% Col. 6+ col. 1 x col. 2, 


63.— Oral Problems. 1. 1, of 10 min.=@ sec. 2 3 sec. in 10 
At sight. min. #times. 38 @ repeated 125 times = 750 
min. 4. Three persons share $12,630 un- 
equally. How much may each receive? If they share equally, 
what must each have ? 
5. $62,500 is separated into # packages containing $125 each. 
6. One of the equal parts of 16,250 is 250. How many more such 
equal parts are there ? 
7, If 208 tickets are distributed one at a time to each of eight 
persons, how many will each have when the tickets are half dis- 
tributed ? 


30 NUMBERS IN REVIEW. 


8, How many bags will hold a million dollars if there are 100 
twenty-dollar gold pieces in each bag ? 

9, Take = of $2.10 from +; of it. 

10. Make two examples: one with quotient abstract; one concrete. 
64. — The Funda- 1, Which of them have to do with combin- 
mental Processes. ing several numbers into one? 2 When is 
the shorter process used? 8 Contrast sub- 
traction and division. How are they alike? 
4, Is the number which equals 10 8’s a sum or a product ? 

5. If you pay $2.38 with a $10 bill, what is left? 6, Four num- 
bers make 87; 12, 16, and 5 are three; the fourth is what? 
7. 87 —15=2 x what? 8 91s what part of 12? 9, 2 wk. = how 
many hours? 10. Why pay 13¢ for 4 yd. at 25¢? For 3 yd. I 


Pye 


For dictation. 


65. — Short Without copying, find the difference : — 
Examples. Ik 2. 3 

Weiner $ 478.36 $ 548.79 379.64 

$ 1399.78 $ 693.78 1633.99 


4. $847.21 — $368.27 5, $2000 — $367.41 6, 932.61 — 878.95 
7. Take the sum of the last three subtrahends from the sum of 
the last three minuends. 
8. If 6 bbl. oil cost $47.70, 29 bbl. cost a. 
9, Multiply 648 by 81. 10, $24.84 x 97, =a. 


66. — Oral Exercise. Supply values of x and y. 


At sight. 

1, yt 3. 4, 
27+43=2 16 SC9e=7 * of 630 =9 x= 7 — 3h 
26--a¢ = 44 Txe=91 Si0 =o xX 2 4.0L (= 
58 —19 =@ 4e¢=—144 15=t of 2x3i=2@ 
GO Sep 2 % =, of T2 6 =~ of 144 e= 7 + 3h 


e+y =100 4 of «= 90 (20 + «= 180 oy Xe 


QUESTIONS AND PROBLEMS. 3) 


5. The divisor is 7, the quotient is 346. How many 7’s are sub- 
tracted from the dividend in finding the quotient ? 

6. In dividing 9000 into 4’s, how many 4’s do we at first subtract 
from the dividend? 7, What is 1 of what remains ? 

8. Compare 6 lb. and 1 lb.; the cost of 6 lb. and the cost of 1 lb. 
6 lb. cost 84, 1 Ib. costs W. of 84¢, or a. 

9, Dis tofex. 27 lb. cost $1.80, 9 lb. cost y. 

10. If 14 lb. cost 84¢, what will 10 lb. cost? 10 x zy of 84 = a. 


a 


67. — Statement of 1. If 16 cd. wood cost $120, 24 ed. cost 
Problems. what? In solving such a problem which of 
Oraland written. these suggestions seem most important ? — 
I. What is to be found out ? (Cost of 24 ed.) 
II. Facts that help to find it. (16 cd. cost $ 120.) 
III. Process, by steps, briefly set down. (24 x p, of $120 = cost 
of 24 ed.) 
IV. Indicated work performed. (24 x +5 of $120 = $180.) 
V. Whether the result is reasonable. (24 ed. should cost 14 times 
as much as 16 cd.) 


Apply the preceding suggestions, and explain orally : — 

2. Bought 12 lb. tea @ 75¢, and 20 lb. coffee @ 40¢. How much 
butter at 30% would cost the same ? 
12 x $0.75 + 20 x $0.40 __ 

$ 0.30 

3, Exchanged a 60-acre farm worth $ 2400 for 200 acres of wood- 
land valued at $13.75 an acre. Gain? 

4, Gave 3000 sq. ft. of 20-cent land for a span of horses and $ 75. 
What were the horses valued at ? 

5, Sixty-four men are employed 25 days in digging a sewer. The 
contract price was $1200. Nothing was gained or lost. What were 
the men paid each per day ? 


Statement. — ~ 


ay NUMBERS IN REVIEW. 


6, A train runs 280 miles in 11 hours. Seven 3-minute stops are 
made, and a hot axle makes a detention of 39 minutes. The rate per 
hour was @ miles. 

7. Six men buy 640 A. @ $125, and sell for $95,000. Each 
man gains w [} of ($95,000 — 640 x $125) =each man’s gain. | 
In the statement what represents the cost of the land? The pro- 
ceeds of the sale? The whole gain ? 

8. Bought 59 bbl. flour (@ $4.75; sold 15 bbl. @ $5, and the 
remainder @ $5.25. Required, my gain. 

9, Three 1-1b. packages will go by mail each for 1 an oz. plus 
registration; by express for 25¢ each. Which way is cheaper ? 

10. A peck, 2 bushels, and 5 quarts are to be divided equally 
among seven persons. Any two receive @ quarts. 


68.— Product of Oral. —1, Explain the process used in each 
Mixed Numbers. example: 3x12=a; 1 of 12=y; 34x12=z. 
2. Give results: 31x12; 81x10; 51x15; 
7 xX 82. 
of 9; $ of 16; 2 of 20; 4 of 28. 
4, Give results: 2 of 45; 7 of 56; $ of 72; #4 of 63. 
5. Give results: 0.6 of 20; 0.8 of 60; 0.9 of 70; 0.08 of 400. 


576 X 


Ino 


3, Give results: < 


No 


8] 6. Supply omissions in the work at the left. 
72 — } of 576 Show what might be omitted in ordinary work. 
504 = § of 576 7. Give directions for each step in multiplying 
4608 =~ x ~ 


by a mixed number. .. 


5112 = 87 x 


Written. — Carefully arrange partial products and results. 


8 98 x 280 1l, 780 x 19,9, 14, 13.5, x 280 
9. 18% x 942 12, 603,32, x 2000 15, 143 ft. x 784 
10. 110%, x 144 13, 18% x 1728 16. 914 Ib. x 1080 


17. Compare 32 with 10. Show quick ways of multiplying by 34, 
334, and 3334. 


QUESTIONS AND PROBLEMS. 33 


69. — Oral Review. 1. Count by ea roe 100 to 0. 2. Count 


en Letitia to 300 by 3873's. 8. To 500 by 622’s. 
4. What two numbers pee than 1 give 7 
as product? 6. Give the sum of 64, 34, 6, 4, 54, 43, 2, 3, 83, 14, 5, 


and 7. 6. Compare the time 6 men need to Sir a road with the 
time required by 2 men. By 15 men. 


7. If I spend 2 of my money and give away $8, I shall have noth- 
ing left. What have I now ? 

8. What is 5 mo. rent of a house hired for $300 a year? 9. One 
year’s interest is $40; 21 years’ will be what ? 


10. Count from 180 to 0 by 18’s. 


70. — Review. 1. tof 4864 = 3. 300 is 1 of a 
At sight. 2. «= of 54180 4, .. minutes = 21 h. 
6. 164 ft.=1rd.; 10rd.=aft. 6. 14 reams = 2x_quires. 
7. Give rapidly the following fractional parts of 100: — 


NUS OS aged SAS Mpeg A ieee Ob ale Se Race 
89 49 8? 22 89 4) 8) 6) 3% 2) 37 6° 


8. If I divide an integer by 356, the largest possible remainder is 
what ? 
9. 2 of 24 is a more than $ of it. (4 of 24) + a= 3 of 24. 
10-210 «0.034, 11. § 2000 is contained a times in $80,000. 
 @=100 x 0.034. 12. wis sj of 100 thousand. 


71.— A Rule of Three 1. If 5 lb. cost 42 ¢,10 lb. cost what? Why 
Applied. is it needless to find the cost of 1 lb.? What 
Oral: at sight. | would 24 lb. cost ? 


2. When 21 lb. cost $3.21, 7 lb. cost aw, and 3 lb. cost y. 
8. 5 for $1.70 makes 35 cost x 4. 9 for $1, 6 for a. 
5. 18%, or 18, of $600 is profit. $36 is what part of the profit ? 


6. $7070 is the value of the crop of a 56-acre market garden 
16 A. at that rate yield $a. 40 A.? 


34 NUMBERS IN REVIEW. 
7, What 42 men can do in a week 7 men could do in WV, and 28 


men in .W.. 


8. Supplies for a regiment of 1000 men would maintain 100 men 
oo or, O00 meno 


9, 10 papers a week, or 2a year. 10, 1000 ft.in 12 sec.; # an hour. 


72.— Problems for 1, Compare 8000 and 2000. Find a short 


Study. way of multiplying 599 by 8000 and divid- 
Written. ing the product by 2000. What is 5955 of 

ia B. CQ. 8000 times a number ? 
257.36 — 129.28 = x 2. Copy the equations at the left, giving 


385.91 — 236.99 = y values to a, y,z Add A; add B. Compare 
536.84 — 327.45 = z the difference of their sums with the sum 
of C, and explain. 


3, In its circuit round the sun the earth traverses about 567 mil- 
lions of miles in a year of 365 days. How many miles a day? 


4, Compare 52 and 364. If 52 bbl. apples cost $175, what will 
364 bbl. cost ? 


5. Find the average weight of 6 men weighing respectively 135, 
176, 180, 138, 207, and 156 lbs. 


6. What would be the duty at 20%, or 4, on 325 bu. beans at 
$1.69 a bushel ? 


7. Mr. Fisk leased an office for 3 yr. @ $374 a month. What 
had the use of it cost him at the end of 21 yr. ? 


8. A barrel of flour fills 8 bags, and costs $4.50. What is the 
gain on 3 bbl. sold at $ 0.624 a bag? 


9, Hans can haul as much sand in 15 d. as Knut can haul in 20 d. 
Which should receive higher day wages? 10. Knut, working with 
a cart and horse, got $60 for 20 days’ work. If Hans had taken 
the job for that amount, what could he have earned a day ? 


QUESTIONS AND PROBLEMS. 85 


73. — The Use of Express in words, giving values to x :— 
Signs. lox b+ T= 2 27+3—-VY/g=2 
Oral. 2 (9+6)x(8—5)=" w=—18—3+./95 


[See pp. 7 and 47.] 3, 57—-5x2=3?+~2 
4, Give name and meaning of each sign just used. 
Why are the results wilike in the upper and lower lines : — 
5. 38+4x5=34200r23 6 6x12—8+2=72 —4 or 68 
(3+ 4) x 5=12 x 5 or 60 6x12—8+2=6x4+2Zo0rl2 
7. When a number has x or + on one side and + or — on the 
other, which process must be performed first? 
8, Compare in value 15 — 5 x 2+ 8, and (18 — 5) x (2+ 8). 
Prove that — 
9, 36+4—45+9=12x3+48-+11. 
10. 2+5x6= twice (2 x 546). 
ll. 4 of (7+14)—1 of 67-9) =8 x 9—3) —48 = (5+ 4)’ 
— (36 + 4)”. 


74.—To Write from 1, 96+12 2. Sq. root of 36 
Dictation. mult. by mult. by 9 
200 + 25 less (7 — 2) 
=4x# = (fc 
8. V36 4, 3 mult. by 5. The sq. root 
x (9 — 5) (6 + 5) of 3 of 48 
= sq. root of less 4 of added to 
(36 — 11) (37 — 10) 2 of 35 
+19 = 30 =a OL 
75. — At Sight. 1, What expressions are here marked to be 


treated as one number ? 
8 x16+8+48 x (146—9)=V3x9x3x108+12—10+1 
2, What signs are used to show that two or more numbers are 
to be treated as one number ? 


36 PRINCIPLES AND PROCESSES. 


3. What number is to be divided by 11 ?— 

6x 544x9)+11 =[6 45) x (0 —4)]+ 11 
Show why it was better to use brackets [ ] than curves (). 
4, w= (12'+24)x V54—5 6 12x44+6x12)+ V100=2 
6. (4800 + 100)+ 0.01 of 6€00=a 7% x=[(7+3) x2—4 of 39} 
8 w=[6 x8—4 x (14+4) + 60] +100 


7 2 ‘ ee ed 
3 le ie: =) + Vor =x 10. [(V25)'— Vex t] + fof 72 =2 


co 


ry 


( 


76. — Substitution The first letters of the alphabet are often 
of Numerical for Lit- used to represent quantities whose value is 
eral Quantities. known. 6a=6 xa; abe=a~xbD xe. 

Oral. SUPPOSe eel a Dee 
Che | CL a eemet 
Find the value of — 


red, One aie ae hl eecaoor 16. db +e VAs 
Die (ce Gf =e 12a ile. sf DO eee 
i at+d $ 
ah af Ue a De LOL 18. 23. d 
C 
4-19 Gao ono aie ee 19, Z 24, de? 
C 
5. 3d 10. 2f—38e 15. abe 20. abce 25, be? 
iE wate Substitute the following values for the letters 


in the problems, and solve them :— 
ai (44 6 39sec =a On Sd == Lose a afi lo: 
1, If } pounds cost a dollars, one pound will cost 2. 
2, What will e yards cost if ¢ yards cost $d? 
3. dyards =ainches. 4 cmiles=afeet. 5. be-+a+ef=2. 
6. Add a hours and e days. 
ff cde 8. fbe 9, a+f+d 


w d c+e 


EQUATIONS. 87 


78. — Equations. Use addition or subtraction in finding the 
Oral. unknown number or quantity represented by 
x, y, or Zz. Explain how you find it. 
l «= 12425 6 42— #2=19 ll, 19+11+4+ 2#=50 
2. 38+ .12= @ Y Fe elf ON 9 12, 724+ 2+14=96 
8 44— 19= @& & 28+ e2=50 13, 404+ 20— a=50 
4, «x=100—72 9 ¢— w«=3 14, $2.75 +4= $ 4.50 
5 a2— 24=—48 10. J lb.—aw2=120z. 15. «—$7.30= $2.84 
16. [ am & years old; in 8 years my age will be 356 years. 
(x + 8 yrs. = 36 yrs.) 

17. After taking $14, $16, and $12 out of a sum of money $3.75 
remained. There were $2 at first. 2—$14— $16 — $12 = $38.75. 

18. A prize cup contains 23 oz. of gold, 10 oz. of silver, and a oz. 
of alloy. The cup weighs 42 ounces. (Make an equation.) 

19, 25 gallons run into a tank, and 46 run out. When the faucets 
were closed, 80 gallons remained. There were 2 gallons in the tank 
when the faucets were opened. (Equation.) 

20. Make a problem about the weather in March to suit this 
equation: 31 d.=12d.+ad.+10d. 


79. — Equations. Use multiplication or division in finding the 


Oral. value of X, y, or z. Hapluin your method. 


Note. —3@ is the same as 3timesa@ dy=5x y. 


meter FO 6.82 =400 11 2 = 8 16s 4e45— 21 


Bey b= or PW Ty = 91. 12: 


I 


Aer iy es, 30 


8. Lofe=16 8142 =700 18, 18, 18y x 10 = 180 


| 


Bese e= 4 9250 =—625 ~-14., 19, 42y+21= 70 


1 


5.144+ec=16 10 44a2= 45 16, 20. tof 16% =120 


| 


-~ 
~ 


Sle 8/E Ele slBaw 
| 
= 
Or 


838 PRINCIPLES AND PROCESSES. 


21, At $3 a day a mason earns $a in a week. (@=6 x $3.) 


22: At 2 cents a mile I can ride a miles for $ 4. ( 2= oe | 


23. My brother is 8 years my senior. This is + of my age. How 
oldam I? My brother is #yearsold. (#w=4x8+8.) 


24, ae =S8m. The distance to the city is required. 
0; 


25, 3 times a number added to7 times it = 280. (8a+ 7% = 280.) 


26. 4 of my money taken from 6 times it leaves $55. (Make an 
equation. ) 


27, 30a = $150; fr =15. (Make problems for these equations.) 


80. — Oral Explain the process; or give the reason. 
Exercises. 1, How many 5’s in half a million? 
For analysis. 2. Divide 80,000 by 4 of itself, and the 


quotient is a. 
3. Compare 2 men’s work with 6 men’s in quantity; in cost. 
4, A board bill for 8 days @ $10.50 a week is $a. 
5. $111 = _.$ Vs. 6. ~ bu. = @ qt. 
7. Compare 62 and 20. If 20 gal. cost $15, 62 gal. costs 1 as 


much or 2. 


8. 900 miles of railroad cost $18,000,000 at the rate of $a a 
mile. $900 is a % of $3600. 


9, Compare the interest of $480 and that of $160. [§ 211.] 
10, Compare 8 mo. interest with 11 yr. interest. 


11. If you know the cost of 7 rape how will sou find the cost 
of 9 of the same kind ? 


12, Find the cost of housing 14,000 Ib. coal @ 25¢ a ton. 
13, At $40 per M., what do I pay for 25,000 ft. of lumber ? 


EXAMPLES. 39 


81.— For Dictation. 


1, Give 6 multiples of 25 

2. 373 + %= 1000 

3. How much fencing is re- 
quired for a 30 ft. square ? 

4, A man leaving his office at 
8 a.m. is absent 20 h. At what 
o'clock does he return ? 


5. Square 2; 3; 4; 5; 6 

Kemduater so 309s 11; 12 

7. What number multiplied 
by itself equals 900 ? 

8. What is ;8, or 6% of 
$ 500 ? 

Umit pe 2230, 1a or 
what ? 


Tato = (0.4 = What? 

1l. Give two exact divisors 
of 52; 51; 57; 58 

12. Divide the cube of 17 by 
its square. 


1S 


18, Divide a million by ten 
thousand. 


14, V36; V49; 121; V81 


15. Give three multiples of — 
fd SG Ds apace oa bys 
16. Give 12 divisors of 144 


17. Make an example in which 
the result is an amount; a re- 
mainder; a product. 


18. The factors of 10a are. 


82.— At Sight. 


1, 2 yd. of 75¢ satin cost 
2. At $0.25, 31 
cost x 
8, 33 yd. cloth @ 162¢ 
4, Beef @ $0.25: 72 Ih. 
cost x 
5, $0.121 raisins: 48 lbs. 
6. Coffee, 95 Ibs., @ $ 0.832 
7. If I strike out from 39 
its largest factor, I divide it by @ 
and get y for quotient. 
8, 22%=a2 V625=y 
9, 123—1728. V1728=-a 
What should be paid for — 
10. 61 yd. of 25¢ ribbon ? 
11. 21 doz. buttons @ 15¢ ? 
12. At$1.00ayard, 2 yd. silk ? 
13. $2 less i of it=2 
14, Explain fis difference be- 
tween 8%, V8, 8 +8 
15. Which represents the cube 
of 3:3.x3, or 8+3+3, or 
Diane Moet o 
16. Find a common factor in 
2oOG mano Le 
17. Which is larger, subtra- 
hend, or remainder in 2148 m. 
— 1075 m.? 
18. Express more simply, 
$0.75, $3 
0.25’ $3 3 


doz. eggs 


40 PRINCIPLES AND PROCESSES. 


83. — Business 1, Go over the computations in the follow- 
Forms, Invoices, ete. ing bill or invoice to find the errors it contains. 


Cuicaco, Aug. 1, 1895. 
Ne. Ifemry dS. Warner 
Bought of FOHN V. FARWELL & CO. 


Nay /8\ 23 yd. Brussels Carpeting, B/,50 \|\84\50 

fune 22| 164 yd. Black Silk, 1.75 \|\/8|98 

Yuly 6| 8& yc. Wamaeutla Cotton, 0 S2; re (5 
08 |\2oO 
Leaw ///0 5|62 
D2 | #/ 

Cr. 
fune 2 vy Caan, $ 25,00 

“BB i Whee 20.00 45\00 
heewved payment, | ogee 


JoHN V. FaRweLL & Co., | 
By Smith. 


Make out bills in proper form. Supply dates and names. 


2, 13 tons Franklin Coal (@ $ 7.25 
6i tons Lackawanna (@ 5.50 
1 Cord Hard Wood @ 11.00 
34 bbl. Cement (QQ 3.25 


3, 2500 ft. Spruce Flooring @ $13.75 uve M.] 
2500 ft. Western Pine (@ 46.50 
1700 ft. Whitewood @ 30.00 “ 
Freight, 8.48 


4, A grocer’s receipts for one week were $365.18; $193.75; 
$96.48; $89.24; $198.65; $479.83. The average of his daily 
expenses was $128.00. What was the cash increase ? 


INVOICES. A 


5. Monday, Jan. 1, 1894, Sam’l Chase had $32.76 to his credit 
ina bank. If he deposited $25 every week-day during the month, 
and $100 extra every Saturday, what amount could he draw against 
Feb. 1? ‘ 

6. In buying 785 music books (@ 85 ¢ a discount of 1 or 20% is 
allowed on cash payments. The net cost is 2. 

7. Bill 62 lb. Formosa Oolong Tea @ 60 ¢ 

30 lb. Maracaibo Coffee @ 241 ¢ 
2 bbl. “ Bridal Veil” Flour @ $ 5.25 
Discount 2% [for cash]. 
8. Bill 371 yd. Dwight Cotton @ 11¢ 
ou 1 yd. Scotch Gingham (@ 25 ¢ 
113 yd. India Silk (Y B1.75 
Credit mdse. returned, $8.75 

9. A firm buys goods billed or invoiced at $1837, less three dis- 
counts. 30% is allowed “to the trade.” After deducting these ;',, 
5% is allowed on “large lots.” The amount due is then lessened 
by 2% “for cash.” The net charge is 2. 

10. Invoice 3% gro. No. 314 Eagle Pencils @ $4.20 
gro. No. 404 Gillott’s Pens @ $ 0.574 
gro. 4to Blank Books (@ $3.66 


rs 


*. 
ee ~~ ( 
Nl 


84.— Oral Review. 1. 2 x 16, or —, is } of 2, 4 of y, + of z. 
At sight. 2. A gross of pens at 4% each costs the 
same as } doz. pins at . ¢ each. 
8. If12 yd. silk cost $36, 72 yd. cost . 72 x qt, of $ 386= 4 of a. 


If > =6, ¢ = what ? 


4. 9x, or 9 x w, = the sum of the 9 digits. 

5. P=; V144=_. 6. The cube of 3 is . W125 = 
1. Add 64, 36, 47, 53, 39, 61, 54, 46, 17, 83. 

8. (64+ 8)+2= V2. 9. Va=648+2. 


10. + of the sum of 5 numbers is their average. Find the average 
of 2, 3, 4, 5, 6, 7, 8. 


492 PRINCIPLES AND PROCESSES. 


85. — Problems. 1. Make an example to show whether it is 
For written work. necessary to begin multiplying with the ones’ 
figure of the multiplier. 
2. What is the value of 795 books at $1.38 each? 8. Write in 
words the second partial product. 
4. If a young man earns $36 amonth, in how many years would 
he earn $ 5670 ? 
5. An importer is charged $45 on a hundred at the custom-house 
on a shipment of goods worth $18,000. What duty does he pay ? 


6. In one minute divide 200,000 by 1728. 7. Write the second 
partial dividend in words. 8. How many such divisors may be 
subtracted from the dividend, leaving a remainder of 9920 ? 

9. A cubic foot of hickory cord-wood weighs 494 lb.; what will 
a cord weigh ? 

10. How many feet inamile? If 10 feet of wire weigh 9 lb., 
what will 2 miles weigh ? 


11. At 15%, how much more than 5000 Ib. of cotton can be 
bought for $800.10? 


1 
86. — Oral Review. 1. + of 15 is z Glee 


Al sight. 2. 40f @=124=—2 of. 
3. The ee on $2 for a year is 12 ¢ or 
6%. The interest for a year on $1 at 6% is what? On $5? $380? 
$100 ? $1000 ? 
4. A year’s interest on $1000 is $60 at 6% ; at 7% it is a. 
5. Give the factors of 28, of 70. Their common factors are 
6. V64 — V64 = Va. 7. #& =64; Ve=2. 
8. 25% of $800 =a 1=what %? 80% of aton=-2 lb. 
9. What fractional part of a dollar is $0.25? $0.331? 12} ¢? 
75% of a dollar ? 162% of a dollar ? 
10. Find the largest common factor of 24, 60, and 100. 
11. 9+5)x3=6e 12 6Gr=945x8 18, 2=V/6i%< 16. 


DENOMINATE NUMBERS. 43 


87. — Denominate 1. Among the tradesmen in your town a 
Numbers. “shilling” is what part of a dollar? What 
Oral. other value is given to a shilling ? 


2. The last decade of the XI Xth century includes the years . to 
... The first decade of the XXth century includes the years 1901 
to .. To which century do the years 1899 and 1900 belong ? 

How many fortnights in a year? Days in a quarter ? 
How many cents = a double eagle ? 

At 4 for 7 cents, 2 doz. cost a, and 4 a gross cost y. 

4 a dozen, 4 a score, and } a gross make how many ? 


IOXAB wf 


If there are 25 envelopes in a package, how many packages are 
needed for a ream of note paper? For 5 quires ? 
8. Ten pencils to a box, how many boxes will a gross fill? <A 
ereat gross of pens fills w boxes. 
9. 5 quires of paper, 96 sheets to a pound, weigh how much ? 
10. A 5-frane piece is worth about $2, or y marks. 


88. — Denominate 1. Change 120000 min. to days. 


Numbers changed to 2. A newspaper prints an edition of 420000 


Larger Units. copies of two sheets. w reams are used. 
Written. 


3. A factory puts up 15000 pint cans of 
[See pp. 8-9. ] 


tomatoes, or bushels. (Allow 4 more.) 

4. In 100000 square inches there are # square feet. 

5. 50000 cubic feet of wood equal x cords. 

6. A number of capitalists control 3,600,000 acres of western 
land. ‘This is the same as # square miles. 

7. 200000 cubic inches make how many cubic feet ? 

8. 100000 ties 3 feet apart will extend how many miles on a 
doubled-tracked railroad ? 

9. 5 million screws are put up in boxes holding a great gross. 
How many boxes are needed? 10. The average weight is 15 oz. to 
the gross; the entire lot will weigh x tons. 


AA PRINCIPLES AND PROCESSES. 


89.— Denominate 1. Thirteen tons of oatmeal will make how 
Numbers changed to many one-pound packages ? 
Smaller Units. 2. 296 bushels = @ pints. 
Written. 3. « pint-bottles may be filled from 728 
[See pp. 8-9.] gallons of vanilla extract. 


4. 6389 tons of baking-powder would cost # dollars, if sold at 42 
cents per pound package. 

5. A school uses 125 reams, or w quires, or y sheets, of paper in 
a year. 

6. One heart-beat a second, or a in a day. 

7. In 7 miles of chain how many inch links ? 

8. 750 rods measure « feet. 

9. 1 acre = 160 sq. rds, ore so.ydi; ory sq.tt.. or 2 sq.m. 


10. 15 cords of stone weigh the same as x cubic inches. 


90. — Problems. 1. January 1, my gas metre read 67500; 
Written. March 31, it read 91500. At $1.60 per 
thousand my quarter’s gas bill is $a. 
2. A. week’s sales of wheat in bushels: 2137, 3476, 972, 3041, 
6732, 1849. Valued at 62% cents. 
3. What did my house cost me as Shown by these items : — 
Cellar, 18 days @ $14.75; mason’s contract, $4575.86; carpenter, 
137 days @ $2.15 and 96 at $3; materials, $576.84; painting, etc., 
397.68. | 
4. Atanauction sale of land the following prices were obtained : — 
3648 ft. at 23¢; 2894 at 311¢; 7642 at 191¢; 8641 at 25¢. The 
auctioneer’s commission was 2 cents on the dollar, and advertising, 
etc., cost $37.50. Required, the net proceeds. 


5. Bought a 100-acre wood lot for $800. Paid 23 men $1.75 fox 
18 days’ work at cutting. Sold 175 cords at $ 2.37, 215 at $4.25, and 
' the remainder with the lot for $800. What did I gain ? 


PROBLEMS, 45 


6. A farmer wintered 17 horses from December 1 to April 1 at 
$12amonth. He paid $28 a ton for 22 tons of hay, and 42 cents 
each for 280 bushels of oats. He had $14 worth of provender left. 
He made $a a month. 

7. I can buy of one firm 732 tons of coal at $4.20 and 75 cords 
of wood at $8.16. Another firm bids $4.16 for the coal and $8.35 
for the wood. Shall I buy of the first or of the second, and save 
how much? 


8. What is an avoirdupois ton of silver worth at 54 cents an 
ounce ? 


9, A city paper prints for a week as follows : — 


Sunday . . . 148,917 Thursday . . 126,839 
Monday. . . 127,832 Friday . . . 128,461 
Tuesday. . . 182,947 Saturday. . . 147,219 


Wednesday. . 147,842 

All but 175,000 daily and 14,760 Sunday papers are sold, the first 
at 2¢ and the others at 5¢ a copy. What is received ? 

10. There are 2741 operatives on a corporation. 12 overseers get 
$5.50 a day, 25 second hands get $2.50, 1505 earn $1.50, 215 men 
and 731 women earn $1.25 each, and the remainder on the average 
receive 96 cents. What is the weekly pay-roll ? 

11. Soundings from a ship’s side are taken every 5 miles over a 
certain course, showing the depth in fathoms as follows: 6, 8, 7, 12, 
20, 21, 19, 30, 40, 80. What is the average depth in feet ? 

12. Find the average receipts of 8 street-car trips as follows: 
$ 3.75, $3.80, $4.20, $2.70, $1.55, $3.90, $2.55, $1.55. How 
many more or how many fewer passengers per trip were carried 
than when $110.55 was received from 35 trips ? 

13. The gates at a crossing are lowered 16 times on Sunday and 
61 times on other days. The keeper’s pay for 3 mo., beginning 
Sunday, March 1, is $99.64, or at the rate of x cents for each 
lowering. 

14. If 3 half-dollars make a pile 1 in. high, how high a pile will 
100,000 make ? 


46 


PRINCIPLES AND PROCESSES. 


91. DEFINITIONS AND SIGNS. 


[FOR REFERENCE.] 


Addition. The process of combin- 


ing numbers, two by two, into one © 


sum. 
Addend. A number to be added 
to another. 

Aliquot Part. 
any number divided by an integer. 

Amount. The result of addition. 

Composite Number. The product 
of integral factors, 1 not included. 

Cube (Number). The product of 
three equal numbers ; the third power 
of a number. 

Cube Root. One of the three equal 
factors forming a third power. 3 x 3 
Xxo.— of Ori. 

Common Factor of two or more 
numbers. A number that is a factor 
of each of them. 

Difference. 


the larger. 

Dividend. A number to be divided. 

Division. The process of separat- 
ing a number into equal parts, or of 
finding how many times one number 
is contained in another. 

Divisor. A number to divide by ; 
it shows how large or how many the 
equal parts of the dividend are to be. 

Equation. Two quantities ex- 
pressed as being equal. 

Exponent or Index. One or more 
figures written above and at the right 
of a number to show how many times 
the number is taken as a factor. 

Exact Divisor. One that gives an 


What must be added | 
to the smaller of two numbers to make | 


The quotient of | 
| bers. 


| 


Factors. Numbers multiplied to- 
gether in making a product ; commonly 
used as meaning integral factors. 

Greatest Common Factor, Divi- 
sor, or Measure. ‘The largest factor 
found in each of two or more num- 


Least Common Multiple of two 
or more numbers. The smallest num- 
ber of which each is a factor. 

Like Numbers have units of the 
same size and kind. 

Minuend. A number to be less- 


~ ened. 


Multiplicand. One of the equal 
numbers to be combined by multipli- 
cation; the factor to be repeated in 
making a product. 

Multiplication. The process of 
combining equal numbers; by repeti- 
tion, into one product. It repeats one 
number ‘‘ many fold.”’ 

Multiplier. The factor that shows 
how many equal numbers are to be 
combined in the product. 

Multiple of a number. 
of which it is a factor. 

Per Cent. Number _ of hun- 
dredths ; units out of a hundred. 

Prime Number. A number with 
no other factor than itself and 1. 


A number 


Product. The result of multipli- 
cation. 
Power. ‘The product of two or 


more equal numbers as factors. 


Quotient. The result of division. 
Reduction. Changing the unit of 


integral quotient, without a remainder. | a number without changing its value. 


RATIO. 


Remainder. What is left when 
part of a number is taken away. 

Root. One of the equal factors 
forming a power. 

Square (Number). ‘The second 
power or the product of a number 
multiplied by itself. 

Square Root. One of the two 
equal factors of a square, or second 
power. 

Subtraction. ‘The process of tak- 
ing part of a number out of it to find 
the remainder ; finding the difference 
between two numbers. 

Subtrahend. <A number 
subtracted from another. 


to be 


92. — Numbers ie 


Compared. 
Oral. 


Compare 7 and 49. 
Compare 8 and 72. 
Compare 42 and 6. 


lin Se 


24 =12 x 2. 
What is the difference between these two 
ways of comparing 24 and 2? 


47 


Signs and Abbreviations. 

() asin (8+ 4) x 5=35 | 
~ asin8+4x5=35 ) 
shows that the numbers enclosed or 
beneath are to be treated as one 
number. 

Vv, the square root of. 

¥/_, the cube root of. 

6%, ré5, -06, or 6 per cent. 

2, 8, as in 5?= the square of 5, or 
25; 43 = the cube of 4, or 64. 

Dr., debtor. 

Cr., creditor. 

G.C.D., greatest common divisor. 

L.C.M., least common multiple. 


Curves or 


Vineulum 


2 = 1, of 24. 


e re, in two ways, 12 and 36; 15 and 60. 
Compare, in two ways, 12 and 56; 15 and 60 


6. Compare 3 in. and 2 ft. 
¢. Compare 2 yd. and 6 in. 
8. Compare 2 lb. and 4 oz. 


9. Why is it that sometimes an integer and sometimes a fraction 
is used in comparing two numbers ? 


10. 16 =+ of x and } of ». 


11. 18 = } of w and s of 54. 
x 


12. With 72 compare 36, 24, 9, 16, 144, 720. 


93. — Ratio. 
Oral. 


The ratio of 6 to 2 is 3. 
The ratio of 2 to 6 is 


~~ 
-_ 
we 


2 
1 wt, 1 
1 2:6=14. 


1. Read these five ways of expressing 


Ratio is expressed 


ratio : — 


as the quotient of one 19 -+94—=—1+2: 12:24=-1:2 
number divided bi , ; 

: J 44=4; 121s to24 as 1 is to 2 
another. 


The ratio of 12 to 24 is 4 


48 PRINCIPLES AND PROCESSES. 


2. Compare 12 with 60. The ratio of 123 to 100 = a. 
3. Which term of the couplet is the dividend? The divisor ? 


4. Mention the <Antecedent, or first term of the couplet. The 
Consequent, or second term. 
5. Compare 0.123 with 0.25 6. Compare 0.75 with 0.124 
81% with 162% $ with 4 
24 with 18 a month with a year 
Compare the end numbers with those between them. 


(182550) 6h. S75 oe ore. 

ail ( t 4 2 d Al 
Hike: 12) U621 . 200 “31L" 1874) 150} at 
(662 2662 414 3662) 


9, 10. 16371400 500 300 43315 


dt 
94. — Oral Review. 1. Give two equal factors of 3600. 
At sight. . 3600 = a. 
2. Find the cost of 23 lb. honey @ 208, 
3 qt. oil @ $0.12 a gal., 12 oz. cheese @ 16¢ a pound. 


3. How much is subtracted by dropping the figure 5 from 205? 
Strike out the factor 5, and what is the quotient ? 


4. How many 7’s from 910 leave 700 ? 


0. tof 210=a 5 0f 210=2a. 31, of 2100 =2. 

6. What is the effect of dropping one cipher from the right of 
an integer? Two ciphers? What is the easiest way of finding 
soo7 Of 21,000 ? 

7. Find the sum: 3600 + 120; 2, of 2400; 1, of 27,000. 


80 ? 900 


8. Find the value of Z of 2400 — 2 of 1500. 


9. Compare 3 and 15. If 15 oranges cost 40 cents, 3 cost u as 
a : 


much, or y. 10. If 3 cost 7 cents, 15 cost w 15 things will cost 
« times as much as 3 things, and . as much as 30 things. 

ll. 25% or + of a thousand bricks are soft burnt. In another lot 
of 21 M. there are five hundred such, In which lot is the proportion 
of soft ones greater ? 


RAPID FIGURING. 49 


95. —— For 1. 2 women receive $2.10 for making but- 
Analysis. ton-holes. One makes 42; the other, 28. How 
shall they divide the money ? 
2. I pay 3 men $36 for shingling a house. One works 6 d., the 
others, 3 d. each. How do they divide the money ? 
3. At the rate of 3 for 7 cents, a dozen eggs cost a. 
If 6 tons cost $24, 24 tons cost a dollars. 
. What will twice as much cost at half as much per ton ? 
. With how few coins can 90 cents be paid ? 
. A’s money is 4 of B’s and 4 of C’s. C has $45; B has a. 
. Divide 3 x19 x 12 by 19. Striking out a factor does what ? 
. 25 is 8 of w and $ of y. | 
10. Divide 8 x 37 x 6 by 37. By 48. By 24. 


SCDONARAA SE 


96.— For Rapid During the year ending June, 1892, the 
Figuring. coinage of the United States was as_ fol- 
Pp. 321 lows : — 
Gold Coins. Silver Coins. 
Double Eagles, 1,086,280 Dollars, 8,029,467 
Eagles, 892,153 Half-Dollars, 1,942,033 
Half-Eagles, 968,191 Quarter-Dollars, 12,093,324 
Quarter-Eagles, 7,561 Dimes, 26,654,641 
l. Find the number of coins for the year. 
2. The value of all the gold pieces coined. 
3. Of all the silver coinage. 
4.im.=~2 ft. 5. 27 acres = a sq. rd. 
4m. = «@ it. 324 A. = x sq. rd. 
4m. = @ ft. 6. 134 cu. ft. = x cu. in. 
im. =~@ ft. 3182 in. = #7 ft. y in. 


7. Begin at the hundreds’ figure to multiply 846 by 327. Is the 
first partial product 2538 or 253,800, and why ? 

8. Multiply 478 by 857. Write in words the third partial 
product. 


50 PRINCIPLES AND PROCESSES. 


9. 50,880 lb. of iron are worth a pound of gold, 16 lb. of silver, 
71 lb. of nickel, or 6560 lb. of lead. How many pounds of iron can 
be bartered for one pound of each of the other metals? Omit frac- 
tions in the quotient. 

10. In the United States silver is produced at the rate of about 
45 cubic inches every minute; how many cubic feet are produced 
on an average in a day ? 

11. A young man with an income of $600 yearly, smokes 3 cigars 
a day, which he buys at the rate of 3 for a quarter. What is 
the annual cost? 


97.— Oral Reviews. 1. When was a man born who, on August 
For dictation. 12, 1797, was just threescore and ten ? 

2. Begin at 13 and give every 15th number to 130. 

3. How many inches in 24 yd.? In 4 of 3} yd.? 

4. A and B get the same wages. A works 3 days, and B 4. They 
earn $21. Find the rate per day. What does each get? 

5. Count by 16’s to 144. By 18’s. By 24’s. 

6. How can you tell the number of desks in your school without 
counting each one ? 


7. What is the difference between rent and interest? What is 
the annual rent of a $3000 house when 10 % is charged ? 


8. If I pay 9 cents for the use of a dollar 1 year, what should I 
pay for the use of it for 4 months, or exe a year? 
x 


9. What is the interest of $300 for a year at 6% ? 
10. Count to 100, 64 at a time. Mention 5 different aliquot parts 
of a thousand. 


98.— At Sight. 
1. Give quickly, 6 times 15, 20, 25, 12, 24. 
2. A rubber stamp printing the words “The Property of the City 
of Brooklyn” costs 4¢ a letter. For the whole, the cost is a. 


DENOMINATE NUMBERS. 51 


3. Fred Jones hires a bicycle @ 25¢ an hour. What does he pay 
for the use of it from 9 a.m. to 3.10 P.M. ? 


4. A ownsa third of a farm, B one-half of it. One has 50 <A. 
more than the other. How much has each ? 


Give results, or value of x. 


5. 25 +10? 6 250+10? 7. 25 +10? 8 18 is Sof 
2.5 x 10? 25x10? 0.25 x 10? (20 + 2) 
95 2257  20+95? - 25 +025? 9. 20—-2—25 —10 


10. Explain the effect of moving the decimal point. 


99. — Denominate 1. A girl is 5, her mother 30. In 25 years 
Numbers. the daughter’s age will be what part of her 
Written. mother’s ? 


2. 1 of a barrel of flour weighs a 44 barrels at $51 are sold in 
1bl. bags at 70¢. What is the profit on 1 barrel ? 

3. $100 = how many pounds sterling? How many francs? 
How many marks? 

4. Give the total value in U.S. money of an English shilling, a 
mark, and a frane. 

5. When lessons are $30 a quarter, the average cost per week is a. 
How many weeks and days in a leap year? 

6. A barrel holding 49 gal. contains # cu. in. How does it com- 
pare in size with a barrel that holds 3 bushels, or y cu. in. ? 

7. Find the profit on an 8-peck barrel of cranberries costing $8 
and retailing a 2 qts. for a quarter. 

8. A ton of wheat contains # bushels. At 21 marks a bushel it is 
worth $ y. 

9. At 60 lb. to the bushel, 3 bushels to the barrel, 9 tons of beans 
will fill # barrels. 

10. How many barrels will be required for 8 T. 8 ewt. sugar, the 

contents of the barrels being 268, 254, 278 lb., and an equal number 
of each size being used ? 


Ve PRINCIPLES AND PROCESSES. 


100.— Cancellation. 1. What is the effect of cancelling or strik- 
Oral. ing out the same factor in both dividend and 
divisor ? 
Illustrate the principle, using — 
2. 100 + 20. 8. 72 + 24. 4. Any larger numbers. 
5. Explain each of the following ways of dividing 48 x 26 by 
3.x 18. 


First Method. Second Method. 
13 48 paee 16 2 16 
x8 x26 89) 1248 AB X78 50 4 8 AB 
39 288 Skee Bx IB 9 
96 78 13|26 
1248 78 382 


6. Which way do you prefer, and why ? 


Choose a method of reckoning mentally : — 
7. (64 x 120)+(8 x 2 x10) 9. (111 x 120)+(40 x 37) 
8. (156 x 75)+(13 x 15) 10. (128 x 85)+(17 x 16) 
1l. State the principle of cancellation. 
12. Divide 320 x 725 x 960 x 570 by 250 x 380 x 144 x 16. 


101. — Written. Solve, using both methods : — 
1. (18 x 19 x 240)+(57 x 36 x 16) 
2. (220 x 231 x 84)+(21 x 11 x 44) 8. 63360 + 5280 
Formulate the following problems ; that is, make an equation showing 
the process of solving each problem. 
Shorten the work by cancelling common factors where possible, and 
Jind the result. 
4. A steamer runs 528 m.in 24h. In 96h. .? 
5. How many periods of 28 h. in 7 wk. ? 
6. 192 T. coal cost $1152; 1536 T. cost how much ? 


7. Give distance in miles between two stations on a double- 
tracked railroad laid with 1056 rails 80 ft. lone. 


REVIEW PROBLEMS. 53 


8. A bag of sage weighs 100 lb. Grinding and sifting costs a 
dollar and wastes 30 lb. of stems. Estimate the value of 10 bags 
after sifting, if sold at the retail price of 10 ¢ a quarter pound. 

9. ‘Two letter-carriers serving a length of 16 blocks, 10 houses to 
a block, on both sides of two streets, deliver in one day 2440 letters, 
760 newspapers, etc.; what is the average number of pieces left at 
each house in one week ? 

10. In a hundredth of the first 5 calendar months how many 
8-hour watches ? 


102. — Review 1. In 1894 the deposits in the Minnesota 
Problems. savings banks amounted to $8,954,575; the 
Written. average to a depositor was $252.63. How 


many depositors were there ? 

2. U.S. post office statistics for 1894: 

No. of post offices, 69,805 Receipts, $ 75,080,479 
Miles of postal routes, 454,746 Expenditures, $84,524,414 

Find average receipts to an office, and the average expenditures per 
mile of postal routes. 

8. Yale’s time in 1894 was 23 min. 47 sec.; course, 4 miles. Feet 
rowed per minute ? 

4. The lowest price reached by wheat in the Chicago market in 1894 
was in September, when it touched 50. The highest was in April, 651. 
I bought 8000 bushels at the highest rate and sold at the mean 
price. What did I lose ? 

5. Compare the mean diameter of the sun, 866,400 miles, with that 
of the earth, 7918 miles. 

6. In the total eclipse of the moon, September, 1895, the moon 
entered the shadow on the 3d at 10:51.7 p.m., and left it at 2: 45.7 
A.M. on the 4th. Duration of the eclipse in minutes ? 

7. Ocean steamers sometimes use 300 T. of coal every 24 hours. 
This is @ pounds per minute. 

8. There are 9 miles of perspiratory ducts in the human body. 
How many ducts are there if each is + inch long ? 


5A FRACTIONS. 


9. The velocity of light is 186,337 miles per second. Light from 
the sun reaches us in 8.3 minutes. What is the sun’s mean distance 
from us ? | 

10. The equatorial circuniference of the earth is about 62 x 3963.296 


. 


miles. What is the length of a degree of longitude at the equator ? 


Fractions. 
103. — Fractional 1, What is a fraction? 2. Is 2 a fraction 
Forms or an integer? 3. In what sense may it be 


Indicate Division. improper to call $ or % fractions? Are they 
mixed numbers? 4 What are they in form ? 
In value? 5. Give other expressions that may be called improper 
fractions. 6. What can you say of the number of units in a proper 
fraction ? 
7. Show how the expression $ ? may mean either 5 coins or 2 coins. 
8. Compare the expressions $ 4, 1 of $9, and $9 +4 
9. Read the following, first as collections of fractional units and 
then as merely fractional forms of indicating division: 48, 27, 84, 10.0, 
500, 10. Give their values in integers. 11. Which A eet: have 
you used as dividends? 12. Which as divisors? 18. Are denomi- 
nators or numerators dividends? 14. Which are divisors ? 


104. — Improper 1. Give the quotients: 21, 43, 77, 3, 14, 20 

Fractions. 2. Give the value of these improper fractions : 

100, £8. 3. How do the terms of improper 

fractions compare in size? 4. What kinds of units are mixed in 
174? In $18.45? 

5. How is an improper fraction changed to an integer or to a 
mixed number? 6, 28-9; 122— 4, 

7. Can 2 be divided by 3? Explain. 8. What 3 equal numbers 
added together make 2? 9. Does 2 mean 2 thirds of 1 thing, or 4 
of 2 things? Explain, using objects. 

10. What is the quotient when the divisor is larger than the divi- 
dend? 11. What part of 3 is 2? 


¥ CHANGED IN FORM. 55 


12. What part of 8 is contained in 5 ? 
Give the value of x in— 


9 Qn . 
Meee ARTA Ibu. 0016, 11.419 — a 


o wv oO 


105.— Mixed Num- 1. How many 7ths equal 1? How many 
bers changed to Im- 7thsind? Howmanyin53? 2. How many 


proper Fractions. 12ths inl? Injof1? In8? In j?- In 
Oral. 85,2 3. Change 82 to an improper fraction. 


4. In changing a mixed number to an im- 
proper fraction (@) what two numbers are multiplied ? 
(b) What two are added ? 
(c) How may your result be proved correct ? 
Change to mixed numbers or improper fractions. Complete each line 
in 20 seconds. 
§. 23,3 


5 
19 
1 


6 ey A Os a265 5 8 29. a CM, Rentoe Eye 
(eeielie? e207 elle) OF Ish 2 62°49 SS oe 110.0 
7. Compare 145 and 1512. 8. 209 = 2 times a. 
OLOF OL0 O01 OOF O:0 29.0 
9. What mixed numbers equal 12°, 109, 190,100, 100, ? 


106.— For Written Change from the fractional to the mixed form, 


Work. and. vice versa :— 
1. 394 8. 49711 5. 51142 7. 4013- 9. 465.27 
2. $9.0 4, 1128 6, 5000 8. 968 % 10. 1899 
11. 29999 gal. = how many gallons ? 


12. How many eggs in 19614 dozen ? 
128 1926 9 5 2 3 6. 
18. Add 1728, 1936, and 23275 
$ 60 nd 22099 $ 3790 
a ees 
15. Which is larger, 1347 or 1429? 
16. Add ;19,, 129, 67 %, 0.97, and 37 %. 
Change to mixed numbers after cancelling : — 
15000 7585 
6408 18. 4773 19, 15990 20. 7585 


14. Find the difference between 


56 FRACTIONS. 


107.— Changes in 1. 3 weeks; 21 days. Compare the size of 
the Number of Frac- the units; the number of the units; the value 
tional Units to corre- of the two expressions. 2. Why is the larger 
spond with Changes number of no greater value than the smaller ? 

in their Size. 3. Give other examples of equal values ex- 
pressed in-units of different size. 

4. Compare the fractions + and ,3, as to size of units, number of 
units, and value. T[llustrate by drawing diagrams, or folding paper. 
5. In the same way compare 3 and 5%. 

6. How does an in- 8. If each term of 
crease in the numera- | 378; be made 4 as large, 
tor of a fraction attect ‘ endl FEE how is the value af- 
its value ? A decrease? J ek Gialstpah We fected ? Why is this? 

multiplied or di- 

7. An increase in | yided by the same 0. iV pach + term or 
the denominator does | number without | ¢ be made 3 times as 
what? A decrease? | changing the value. | large, how is the value 
Illustrate with 4. affected? Explain. 


10. What is done to the fractional unit when the denominator is 
doubled? When it is halved ? 

1l. What is done to a fraction when its numerator is doubled ? 
When it is halved? 12. How do you tell whether a fraction is 
large or small ? | 

13. Show the effect produced in a quotient by multiplying or 
dividing dividend and divisor by 5. 


108.— The Terms 1. Applying the principle stated above, 
of a Fraction made how may a fraction be expressed in larger 
Larger or Smaller. terms? 2 How in smaller? 8. How do 

| you change 3ds to 12ths? 12ths to 3ds? 
4. How many 12ths in 2, 3, 3? 

5. In changing 4 to 20ths why do you multiply both terms by 4 
instead of by 5? 6. In changing #8 do you divide both terms by 6 
or by 7, and why? 


REDUCTION. 57 


lod | m & 9 
vf Change to AOths: oe ak > st 40 24 50. Balt 160 
SS 16 “be 30" 160° 200: -20° 240 


Par e.. Goo es see 9 Peer ied @ ia, lt 10 


- 


— —» ——9 ——=» Lp, toe . Sy ahr ees eae, 01 trem le erer el ere 
21 49 28 42 84 91 CUneesUeriew ta oe) b 

10 fete “Oooo LU ae 0h ae 80. 100 

th laps Eiewer | mrs ? ’ eas 
few oo) 40) te. 9 ee ae 00g re  or 

11. Without changing the value, make the denominators of these 

et) 4 BR. 4 1 8 Et 102 3 1 
fractions 156: 4, 24, 38, 44, 42%, ade 4 7iv 

12. In changing a fraction to lower terms, how do you affect the 
size of the fractional units? Their number ? 


o 
= 
‘ 


“ 
| 
Ne 


5 
8 


13. Why does multiplying both terms of 
unchanged ? 


@ by 8 leave the value 


109. — Rapid Give an equivalent fraction in larger or 
Changes. smallest terms : — 


At sight. 


| ae g 21 64 27 36 73 

"4 56 y . Tahewie = 06. 108-505 
Plat 568 94 Of : 

2. on 7 = 7. 75%, 0.50, 80%, 0.90 


4_w 5_y qty ee aye (Oped ye 
5 45° 9 45 ~~ 105" 7 500 yy «20 
alae 2 v 9 
4. = we 12 to 48ths 9, Lies x — 105 _ : ao : 
16 24 96 ae AO ae ear) are 


5 48 3 96 16 12 2  »v 288 


.—~ -——», — to 60ths 10. ——= = = ___ = 
12 120 4 144. es-y~. 24 288)" u 
110.— Greatest = = 1. Why do we call 72 a composite number 


Common Divisor of and 75 a prime number? 2 What com- 
Two or More Num- mon divisors have 36 and 108? 60 and 90? 
bers. 3. What is their greatest common divisor 


(g.c.d.) ? 


58 FRACTIONS. 


4. Numbers without a common divisor are prime to each other. 
Explain which of these are prime to each other: —27 and 35; 
16 and 45; 27 and 45. 

5. What advantage is there in using the g.c.d. in changing fractions 
to smallest terms? 6. Change to smallest terms, using the g.c.d.: 


2500108 69 Ph 20 6 
Tes) TH, $2, 80%, 0.125, 4%. 


111. — Finding For small divisors. 1-8. Which of these 
G.C.D. when not numbers are multiples of 2? of 3? 4? 5? 6? 
readily seen: Sno aoe OLE | 
360 Dige 8397 2160 
1728 6984. 6624 3240 


Notre. — Any number is a multiple 

Of 2, if it ends in 2, 4, 6, 8, or 0; i.e. when it is even ; 

Of 3, if 3 divides the sum of its digits ; 

Of 4, if the last two figures are zeros or express a multiple of 4; 
Of 5, ifit ends in 0 or 5; 

Of 6, ifit is even and divisible by 3 ; 

Of 8, if the last three figures are zeros or express a multiple of 8 ; 
Of 9, if 9 divides the sum of its digits ; 

Of 10, if it ends in 0. 


For large divisors. 9. Using 8 and 12, or any other two numbers 
having a common divisor, show that — 
Any divisor of two numbers is a divisor — 
I. Of their sum. Il. Of their difference. 
Ill. Of any multiple of either. 


To change 244 to smallest terms, applying the principles just 


stated. 
10. Dividing the greater number by the less and 


247)533(2 the last divisor by the remainder, we find 13 to be 
494 the g.c.d. Show — 
39)247 (6 a. That any divisor of 247 and 533 must 
234 divide every 247 in 538 (III.), and also the 
g.c.d. = 13)39(3 remainder 39 Clays 
39 b. That, therefore, the largest common divisor 


cannot be more than 39; 


ADDED. 59 


c. That any divisor of 247 and 39 must divide every 39 in 247 
(IIL.), and the remainder 13 (IL.); 
d. That 18 divides both, but no larger number would. Hence 13 


9 53: 18 | 247 — 19 
is the g.c.d. of 247 and 533. 13 | 247 = 18. 


ry 
i“ 
~~ 


> 
fr) 
=) 
bo 
— 


112. — Fractions a. 10. 


ols 
saalco 


205 6 9_ 1 
2 171 465 1325 
to Smallest Terms. 9 136 261 8. 333 1], 1281 
fT 290 ee aoie | Se SOG 
ritten. 278 209 123 1656 
Written 38. 353 6. 333 9. 333 12. 3335 
113.— Like Frac- 1. What are hke numbers ? Give examples. 
tions to Add and 2. What is an integer? A fraction? Isa 
Subtract. fraction a number? 3. Show the difference 

Oral. between integral and fractional units. 


4. With regard to each of the follow- 
ing numbers mention.(a@) the integral unit, 
(6) the size and kind of the fractional 
unit, (¢) the number of fractional units : — 
$ pk.; 7% yr.; 48; $3; 7 m.; $0.15; 6% 
of a day. 

5. Which of the following fractions have units (a) of the same 
size? (b) Of the same kind? (c) Of the same size and kind? 
Which are like fractions? #yr.;4yr.;¢yd.; 7 day; $4; 3 yd. 

6. Why not add 7 1lb. and $3}? #2 wk. and i wk.? 


Like fractions 


have units of the 
same size and kind. 


7 pee AN e 147 +21 7 % . 
7 —+—+—=— 49. ——=— ll. 72% — = 0.48 
271272 1D 36-36-36 2h — 599 
fue eee a. =e 050) 19) 16212188 = 2 
64 64 64 64 
13. How are like fractions added? How subtracted ? 
114. — Unlike 1. 14d.4+3 wk. =2 wk. or y days. 
Fractions to Add 2. Mention several unlike fractions. Why do 
and Subtract. you call them unlike ? 
Oral. 3. 4,3,4,3. Which of these fractions have a 


common numerator? A common denominator ? 
Which are like fractions ? Which unlike ? 


60 


son Suede Siu sige gaan 


the like fractions first. 
Why change 18ths to 6ths 


Why change 8ths to 24ths ? 

7. Which term is common to like frac- 
tions? Change 2 and ;§& to a common 
denominator. To the least common denom- 


inator. To 20ths. 


FRACTIONS. 


=a ed 
6£§ t+i1=2 7 a 
40 Add Ly 03 Before finding 


their sum or differ- 
ence, fractions must 
be changed to like 
fractions. 


8. In adding 3 and 3} shall we use 4ths or 60ths as the common 
fractional unit? 4 or 60 as the common denominator? 9. Why 
is it easier to use the largest common units — or the smallest common 
denominator —in adding? 10. Give three steps in adding 7 and 5%. 


115. — Oral or Written. 


. Change to a common un 


i 
2 3 
. Add or subtract numerators. ye 5 aes 
, 4 

4 


1 
2 
53. Simplify the result. 
4. Find short methods. 


9. 
10. $+ 4+de+at 
ll. 20% +4+4+0.20 + 
1. F+E4+ E4404 
13, 4 haere 
14. 18% +0124 3+4 
20. + wk. + 325 yr. + 


116. — Multiples: 


2o+s Oa 
ae tet 3 6. $—-@ 

100 le er 

#+3 8. 25% + $F 

fo aecte aes 
75 lo. ¢typete—ats 
$ 16. § +43 + 36 + 23 
25% 1% 34-2484 
18. 36% —4+4+0.14—4 

: _ Wee +e-Ht a 
id. — 5k mo. + 180 min. = « 


1. Show that 36, 60, 72, and 120 are multi- 


Common Multiples; ples of 12. 


Least Common Mul- 


2. Show that 50, 60, 80, and 120 are multi- 


tiples. ples of 10. 


ae 


Norr. — The term “‘ divi- 
dend’’ may be substituted for 
‘*multiple.”” 


3. Show that 60 and 120 are common multi- 


ples of 10 and 12. 4. Show that 60 is the 


least common multiple (1.c.m.) of 10 and 12. 


LEAST COMMON MULTIPLE. 61 


5. 2, 6, 3, 10, 15, 5 are factors of 30. Which of them are the 
prime factors of 30? . What is the product of these prime factors ? | 

6. 2, 6, 3, 14, 7, 21 are factors of 42. Select from them the prime 
factors of 42 and give their product. 

7. The product of the prime factors of a number is always 

8. Show that any multiple of 30 contains all its prime factors. 

9. Show that any multiple of 42 contains all its prime factors. 

10. Show that any common multiple of 30 and 42 contains all the 

prime factors of each. 


117. — Finding the To find the least com- 80=2x3x5 

L.C.M. mon multiple of 80 and 42=2 x3 x7 
Aero et Hemerimilgiplartic Ne tate ee ee 
containing only such prime factors as are 
needed to produce each number separately. 

1. Will a number whose factors are 2, 3, 5, 7 be a multiple of 
2x3x5 or 30? Of 2x3x7T7 or 42? 2. Why need not the 2 
and the 3 be used twice in this multiple ? 

3. What factor not found in 42 is needed in the common multiple ? 

4. Find the l.c.m. of 60 and 84. 

5. Find the least number exactly divisible by 60, 72, and 108. 


Oral and written. 


6h Fo & eG 6. What prime factor of 60 is not found 
72=2x2x8x8x2 among the prime factors of 108? Of 72? 

108 =2x2x3x3x3 7. What is meant by the least common 
5x 2x 108=« multiple of several numbers ? 


8. What is the l.c.m. of 60, 84, and 132 ? 
9. Find the l.c.m. of 45, 90, 100, and 200. 
10. Is a multiple of 90 a multiple of 45? 


45 . : 

ay ee eae Compare 200 and 100 in this respect. How 
100 then may the process be shortened ? 
200=2x5x2x2x5 Nore 1.— The prime factors of large numbers may often be got 
200 XX B= 2, by finding composite factors first, and then the prime factors of 


these. Thus:— 
120 100 12 eX O) Ke Mie OCS) LOU LO xe LO = (SO) 0 (YIX'S %.B), 


Norte 2.— For other methods of finding the ].e.m. see the Appendix, p. 8, 


62 FRACTIONS. 


Find the least common multiple of — 


al 15 2145 
12.16, 18, 27, 72 


13. 16, 25, 80, 100 
14. 12, 18, 96, 144 


15. 34, 85, 51, 68 
16. 480, 600, 1000 


Practice in Changing Fractional Forms. 


118; — Oral. 


l. Read as mixed numbers: 
39 47 289 365° 300 
29° 


CE) ye) Sey BR 
2 Putinte Sa he form: 
6 F&F 6 4 
45, 58, 102, 20%, 62, 134. 
3. Change to ecatteat terms: 
2'8 V4 6:85 325 64596 09 1.0'S 
44). 51? 72? 72? GE 84) 120° 
4. Simplify the form: 
JA AMO EE op Ete i i hen Ss 
Ci mle t oe LAUT 0.160. 
5. Read as 144ths: 1, 2, 54, 
Si TE ay A SOD) 
167 167 7.27 2:38:87? 14402 Teale 


6. Change to smallest terms, 
} - 25 2 52 48 
using g.¢.d.; ee eb 36 30 


32 Olea? 72 
480? 108 288? TET 


7. Find l.c.m. of 5, 7, 35, 70; 
8, 16, 128. 
8. 64 = how many 8ths ? 


3 2 2 @ 


Ope ean pee 
79-66 2 eee 63 
Di tn0 Ate 
0 0 6 12 8° 9 


11. What parts of 144 are 1, 2, 


Beess, OMe. 729 


12. Compare in value: 


Pee Wey ee Myer ary eee 
695) 175) 98? 99) 56) 6S 


18. Read in order of size: 


By Sb wie Be eke, leh. 3! 12 
TiO? oAmige iso) wlelvedeo? sel eisai 


119. — Written. 


1. Write two large fractions, 
using four figures. 


2. Express the same values 
with 12 figures. 

3. Change .to mixed num- 
bers: 8248; 1990; 700, $251 

4. Simplify the ae of $139 
486 doz.; 352% ed. 

5. Change 75% 
terms. 


209 


to smallest 


as 625ths. 
1. 52%, in smallest terms =? 
8. Change 27 and 45, to like 
fractions. 


18 
6. Express 48 


9. Find the least common 

multiple of 16, 48, 96, 108. 

10. Change to smallest terms, 
Sa CETE 

3080 

1l. Express 24% with four 
figures. 

12. Condense this expression : 


13. Shier ae 19613 and 2234. 
14. What part of i7 28 is 576? 
63 x 2000 x 31 


te, ete 
125 x 1260x70 


120. — Fractions 


Added or Subtracted. adding or subtracting 


Process. 
15=3~x 5 
18=2x3sx38 
sO = Py 1, CO: 


or l.c.d. 
6 
LL 90 _ 66 
13 °° 90° 90 
5 
7 _ 36 
13 °° 90 90 
Sum = 49) = 143 


121.— Written. 


ADDED OR SUBTRACTED. 


63 


1. What is the first step to be taken in 
it and 54? 2. Is the 
common fractional unit readily seen ? 

38. What is the lem. of 15 and 18? 
4. What then will be the smallest common 
fractional unit, or the least common denom- 
inator ? 

5. How many 90ths = 
how many 90ths? 54? 

7. How are the size and the number of 
fractional units changed by multiplying both 
terms of 14 by 6? 


? 


Sag eS wl. a pe Dee 
Lioiee Lidwe ~ Lice 


8. What remains to be done after the 
fractions have been changed to like fractions ? 


l ~+A=2 11. Add 7%.+ 44 to the sum of $4 and 41. 
2. $f+ii=a 12. What shall be added to 5% to make 49? 
8. Fe t+e=H 13. Which is greater, °, or; ? How much? 
4, 294 298=¢% 14. Add 23 to 4%. What should be the 
5. 38; + 38255 = a first step here; and why ? 
6. 0.8—0.625 =a 16. 2444 2314 723+ 16541818 Add 
7. 80% —3=2 the fractions mentally. 
8 $4 5428 = 16. Add 138, 16%, 0.027, and ;48,. 
9. 518 + 2149 = 17. 323 4+1714=¢2 

10. «7+ 13 = 729 


122. — Subtracting 
Mixed Numbers. 


Oral. 
Process 
W575 = 1535 = T44P 
5711 = 5733 = 5733 
Difference = 17$4 


1. In subtraction of integers, what 1s done 
when a digit in the subtrahend is larger than 
the one above it in the minuend ? 


2. How is the 4%, 
at the left? The 74? 


3. Tell how to take 73 from 154. 


109 obtained in the process 


64 FRACTIONS. 


4.123—-Ti=a 6. 10—3§— 25 8. 98 — 514 
5. 100 —aw= 83 1. 52 — 33 — 3 9. 334 — 121 
10. 20 —{—{-— {4-4 12. 15 — 28 — 28 — 26 
ll. 93 — ¢ — 5,3, 18. 203 — 3 —75% — 0.75 
123. — Written. 1. 6313 — 417 3. 725 — 1514 
2. 944 — 18,3, 4, 1913 —517 
. 14,9, — 6.51 6. 0.875 — 0.625 +, By + hts 


. What is the differ ence between 874 % and 662 %? 
. What added to 17,9 gives 29%, 
10. From 184 take 9.935 


5 

7. 11411 — 1632 + 8419 — 1623 
8 

9 


= 


124. — Problems 1. 4, 2, and + of a number=110. Is the 
with Fractions. unknown Aine larger or smaller ? 
2. A barrelis $full. Draw off 4 of a barrel 
and 2 of a barrel. What part remains ? 

3. A stone wall cost $1 arod. What costs 3 days’ work, or 63 rds., 
53 rds., and 72 rds. ? 

4. What % of an income is collected when 0.125 of it, 25%, 0.35, 
and 0.025 are paid in ? 

5. A chimney contains 182 courses of brick. =; are under ground, 
24 roofed in; how many courses are ara ? 

6. How een cords of wood in 2% ed. sawed by hand, {24 ed. by 
machine, and 7% cd. chopped ? 

1. Two dane contribute + and 1 toward filling a reservoir, 
springs contribute 4+ and 4, surface ma the rest. How much more 
do the pumps yield than other sources ? 

8. Ina10 acre marsh lot three men cut 4, =3,, and 2 of the whole. 
What part remains oF a fourth man to cut ? f 

9. If you Bn eh , of what you have in one way and 3 in another, 
what remains ? 

10. At $3 a day what is due a man for working half a day, 2 d., 
$d., and 213 d.? 


Oral or written. 


MULTIPLIED. 65 


125.— To Multiply 1. 9x7 units =a units. Does it matter 
a Fraction by in- whether these units are integral or fractional ? 
creasing the Number 2. Then it follows that 8x3 fifths =@ 


ee fifths, and 9 x gigs, 
SOLS 
A : = - or 3 and 7. The product of 22° by T=a 
4. 6x35 =2, ory andt+ 8. 24x $2=a 
6. §4+8+4+8+4848=% 9, 100x $4— 4 
6. ;4 multiplied by 11 = a 10. 50x $8 =a 
1 23 a 12. Multiplicand 54, multiplier 9, product ? 


13. In the preceding exercises, have we changed the number or 
the size of the fractional units? 


126.— To Multiply l. James has $3, Henry has twice as 
a Fraction by Increas- much. Has Henry 6 quarters or 3 halves? 
ing the Size of Parts. 2 x }=—3or $? . 
2. If I have ten }’s of a dollar and change 
them for as many coins of double the size, what fractional parts do 
I get, and what increase in value ? 


3. Compare ;, andi. 4x i =- lb. What change is made 


t 
4 
‘ | 

in value when instead of 16ths, we take as many 4ths ? 


ind the product by increasing the size of the parts : — 


4,.6xq 7. 10 x 34 10. 15 x 4 
5. 8 x 13 8. 12 x 48 yd. ll. 25 x 348; 
6. 83xi 9. 18 x 4 m. 12. 36 x 7; 
127.— Multiplier, 1. Which is larger, product or multiplicand ? 
a Fraction; Multipli- 2. Is it proper to say either 4 times $6, or 
cand, an Integer. $6 multiplied by 4? 3. What about the 


product here? Is anything really multiplied ? 


66 FRACTIONS. 


4. Infinding } of $8 to be $2, do we multiply or divide? 5. 2 of 
6 yr. (or 2 x dof 6 yr.) is 4 years. What part of the multiplicand 
has been taken? What does multiplying mean ? 

6. Show that multiplying by a fraction is finding one or more of 
the equal parts of anumber. 7. Is this more like multiplication or 
division ? 

8. 2of 24h. 13. $ of 830=5 x 42 = 20139 = 218. Explain. 


9 tof 36in, 14 2 of 3) oe ae = 21f. Explain. 
( 

10. 14 of 84mo. 15. % of 20 18. =3, of 100 

ll. fof 60d. 16. 3, of 50 19. 0.23 of 200 

12. 42 0f $100 17. Sof 100 20. 24% of 500 


128.— The Productof 1. Show that 7, of $20 =20 x $54 
Fractions and Integers. 2. What is the a of series 


Last 3. Compare 4 not 106 = = x with 105 ae 
pp 5 

ee au 
4. — of 144=—-—=404. Explain. 

60) 

5 
5. 13 of 105. 7. 18% of 250. 9. 17 of 225 tons. 
6. 65 x +5. 8. 8 of 85. 10. 840 x 44 yd. 


129.-— One Factor 1. At $3.50 a yard what will 2 yard cost ? 
a Mixed Number. 2. If one revolution of a an requires 
Written. & of a second, how long will it take it to 
revolve 1000 times ? 
38. 20f 400 =a; 16 x 400 = y; 162 x 400 =2+ ¥. 
4.17 x8=2; 17 x 300=y; 17 x 3008 = a+ y. 
5. At $0.95 a pound what will 742 pounds of tea cost ? 


PROBLEMS. 67 


Process. 
$0.99 Copy the accompanying process, supplying 
_ 145 values for the letters. 
9) $6.65 or a x $0.95 
0.738 = § of b Nore. — When the cost includes a fraction of § cent or more, it is 
3.80 =c x $0.95 customary to count the fraction as another cent. Why is this done? 


Answers given in this book conform to this custom. 


66.50 =d x $0.95 
$71.03§ or 6. Price $8.75; quantity 183 cords; total 
$71.04 RE 

7. Weight of one bag of coffee, 352 lbs.; 19 bags weigh what ? 

In the process at the left give the values of 


Process. a, b, ¢, and d. 
355 Ib. 8. 15 boxes; one weighs 183 lb.; all? 
Doe 9. $6.25 each; 273 yards; cost? 
8)133 or 19 x a 10. 15,4 miles an hour; 25 hours ? 
163 Ib. 19 x b 11. Rent, $28; time, 7; months. 


alte, Coe : rat : 
12. Time, 2911 d.; sailing rate, 185 miles. 


30 Od Ko) : : 
13. 2 of $785 was lost in speculation; 
681g + ataey 
what remained ? 
130.— Problems. 1. What remains of a 49-yd. piece of cloth 


For written work. after selling $ of it, 3 of the rest, and 4 yd.? 
What is the remnant worth at 83¢ per yard ? 
2. Another piece of 47 yards is damaged. One half sold at 7¢. 
Of the other half 23 yards were unsalable, but the rest went at 5¢. 
Give the total receipts. 
3. At the rate of $12 a day, figure a board bill in dollars and 
cents for 3 months from August 1. 
4. If a glacier moves uniformly a hundred feet a year, how far 
does it go in 181 days ? 
5. A sawyer works up wood at the rate of {9 cd. a day. What 
can he do in 26 weeks if he takes a half-holiday each Saturday ? 
6. A wind storm passes over ,), m. in 3,3, sec. In what time 
would it go a mile? 7. If it travels +3, m. each second, how far 
would it go in an hour? 


68 FRACTIONS. 


8. Find the cost of seven 50-gal. barrels of oil at three for $16.71. 

9. Supposing an empty barrel to be worth $1.25, what is the oil 
worth per gallon ? 

10. An unfailing spring flows 374 barrels daily. How much 

would it yield in October ? 


131.—To Find One 1. 2 of 6 things (apples, dollars, fourths) 
or More Equal Parts =v. 
y) y) 
oe er or 2. = of 10 twelfths =a; = of e a = ses 
D Og Lele 
ans yp IP as 
SUS ore a 5. of 22 =a 1. ps of 38 =a 
4. tofis§=—-2x 6. 75 of 42 =a 8. tt of 72 =a 


To separate * into 5 equal parts and give the value of 3 such parts ; 
that is, to find 3 of #, or to multiply 2 by 2. 


Process. 9. How does increasing the denominator affect the 
lof3=,3, size of the fractional units? 10. If we make the denom- 
of }=,% inator 5 times as large, how is the value of the fraction 
: : ; = s changed? tot#—. ll? If of a-traction=— 3. oF 
2) 


it will be how many times 53, ? 

12. Which part of the process gives the product of the numera- 
tors? Of the denominators? 138. Make a rule for finding the 
product of two fractions (i.e. for finding one or more of the equal 
parts of a fraction). 


132. — Fractions 1. Find 19 of 24, or multiply 24 by 19. 
Multiplied. 2. Of what use is cancel- Process. 
Cancellation. lation? 8. On what prin- 12% 3$=220=7 
ciple is it based? 4. Which a i h 
is easier in the process at the right, to change the 12° 338.7 
product to lowest terms or to cancel first ? ee 


5. ay of #9 THX Hs t 
6. +4, of 32 8. 28 x 24 10. 96% of 23 


MULTIPLYING FRACTIONAL NUMBERS. 69 


133.— Multiplying 1. Find the product of 14 and 24 


i 


Fractional 2. What is 2 of 74 (2 of 43)? 
eos 8. 88x 63—-a (First step ?) 
Written. 4. How may the product of small mixed 
numbers be found ? 
6, 22 x 73% 1. 4 of 8 of 34 9. 16% of 54 
6. 44 x 158 8. 22 x 44 x Th 10. 94 x 1219 
11. 168 x 246 = what? [§ 68.] 12. 68 x 1373 = what? 
i aie, 13. 84 x 126 Explain the process. 
xplain the process. “ 
¥ 14. 149 x 72 187% 
246 ; $ 63 
65 15. 83 x 2094 : 
6 x 137 = 822 
16. 68 x 1943 
ee \=(5 x 246) +8 Ne Sar 8 x 137 = 1028 
163} 17. 264 x 7952 6xi = 33 
P10 OX DAG Fe 15 
‘ 18. 84% x 641,5, x' = # 
3 ; 
ice 19. 962 x 109.8, 92834 
134.— Business Make out bills in full for — 
Problems. 1. 9. 
For written work. 17 doz..@ $ 1.624 178 yd. .@10¢ 
152 doz. ..@ $ 1.00 13 yd. —.@ 621 


3. Find the cost of 92 tons of coal at $ 7.41. 


4. ‘Twenty pounds of sugar bought @4,% are sold for $1.25. 
At this rate what is gained on a barrel of 200 lbs. ? 


5. Oil is bought at $3.50 for a 42-gal. barrel and retailed at 121¥. 
The gain is what part of the cost? 


6. Oranges bought at 3 for 5¢ are sold at 4 for 9¢. What is 
gained on a box of 9 doz., 1 in 12 of which are worthless ? 


7. I can buy blank books of one dealer at the rate of $1.25 a 
hundred; of another at $1.60 a gross. How much is one offer 
better than the other ? 


70 FRACTIONS. 


135. — Products at Multiply each fraction in the table by the 


Sight. number at the end of its line or column. 
Oral. Change any fraction in the product to a 


lower eter et when possible. Thus :— 
xo yd. = 20 yd. = 22 yd. = 2 yd. 8 in. 


vu 


10 


& sq. yd. 


8 : 
15 M1. 


Division of Fractions. 


136.—Fractions to %+5. To divide % into 5 equal parts. 


Bea ake 1. Each part will be t of 3 or — 8 
[See §§ 107, 131. ] 2. What is meant Dads § 42 
4. Divide 2 ft. into 6 equal igen 5 x “+ @ = ah. 
6. 16+4= a 7. 43+4= a 8. Using the two preceding 


examples as illustrations, give two ways of dividing a fraction into 
equal parts. 9. When is the second method used? 10. Which is 
shorter ? 


ll. 4 of 24, 14. 51, of 4 17. 0.15 +6 
(Dy yet: Woe LE aby 18. 48% + 16 
1gaaeees 16. 4, + 20 Vid eee erin 


DIVIDED. 71 


137. — The Divisor 5+3. To find how many times 2 is con- 
a Fraction; the Divi- tained in 5. 
dend an Integer. 1. How many times is } contained in 1, or 


how many 4ths in 1? 


OP od 


a eS 5 x 
2. How many 4thsin 5? 5= 7k 8. 3’sin20? 249+2=2. 
4. 2ft.+3in.=2; or, since dividend and divisor must represent 
y] ? 
like units, 24 in. + 3 in. = @. 
a eect: Geay ie ee 8 j » t] 
5. 72 1. T+2=a. 9. 58 in 8, x times. 
6. 8+4=2. 8. 2 in 7, # times. 10. Divide 17 by 3. 


—J 


138.— The Dividend #+2. To find how many times % is con- 
and Divisor both Frac- (tained in 3. 


tional. 1. Which is the larger, 2 or 2? Are the 
: ay OA se ; 
units of the same size? 2. ~=—; -=>~3 w+y=11. 


Oe Be 
3. 4 or = is contained how many times in ¢ or 12? 12+5=2a. 
4, +2 6. §+32 8. 55, in $ 10. Divide 22 by 55 
5. $m 7. +e 9. $+F 11. Divide 0.9 by 30% 
12. What is the first step in dividing days by hours? In dividing 
4ths by 5ths? One fraction by another ? 


139. — Denominate 1. 2 of a case of slates at $43 a case cost 
Numbers. $a. 2. A ten-pound box of marbles contains 
Drat: 4 yellow, 2 blue, and 48 red. The box con- 
tains how many? 8. The box cost 30 ¢, and 

the marbles sell at ten for a cent. The profit is a. 
4. A girl who earned 2 of a dollar gave 3 of it to one who had 
nothing. With the rest she bought three things that may have cost 


x, y, and z cents. 

5. A dealer bought $10 worth of oranges. After selling a fifth of 
them for $3, he sells the rest for what the whole had cost. What 
was the profit ? 


ke FRACTIONS. 


6. 1 of a bushel of berries are picked; 4} of them are sold to one 
man, 1 of the remainder to another. How many quarts are unsold ? 


7. A certain sand-glass runs ten minutes. It runs out twice during 
© of an hour are spent in the morning, and 


practice time at noon; 
In 6 days it would be a. 


jz at night. Give the sum in hours. 
8. At 51¢ each 64¢ buys a, and 80 ¢ buys y. 
9. $10,000 in postage stamps are divided among 4 offices. If 3 
of them are twos and the rest ones, how many ones does each office 
get? 10. How many twos are distributed ? 


140. — The Divisor (a) +3 To find how many times 3 is 
Inverted. contained in + 
Oral and Written. First Process. 


1. To what common unit are the fractions changed ? 
2. How many times are 24 units contained in 35 units ? 


Second Process. Analysis of Process. + is contained 
in 1 five Pens 3 is contained in 1, 


3 : ; 
4 of 5 times or 3 times. Since 3 is con- 
at 1 it will be aon < of 3 times or 


ala 


tained in 1 3 times, in 
3 times = 141 times. 
But 3 is the divisor 3 inverted. Hence to divide one fraction by 


another, we may 
Multiply the dividend by the divisor inverted. 


3. What advantage has the second process over the first ? 
4. What disadvantage may it possibly have ? 


Apply the shorter process to the following and explain it : — 


5. #+2 7 $ini 9. What part of 3 in 3? 
6. ~§+¢4 8. £in 54 10. What part of 5% in 3? 
Notre. — Cancellation will often shorten the process in division as in multiplication of fractions. 
25. 2) pei a PR, es eB es 6 340 + 
11. Tt eatee BR) 12. 44° 71 18. 250 ° %5 14. 625 > as 


DIVIDED. 73 


141.— Division of 1. Divide 123 by 32. (Change mixed num- 
Mixed Numbers, bers to improper fractions.) 
Complex Fractions. 2. 158 + 93 5. 822 + 322 
Written. 3. 733 + 574 6. 1000 + 662 
799 - A498 321 
24. 3 
8, ~2 1s a way of indicating the division.of 2} by 73. 2=~. 
i> = 
3 


Such expressions are called complex fractions. To ahenne them 
to simple fractions, multiply each term by a common denominator 
of the fractions. Thus:— 


7k x6 44 
2 3 ue 3 
_ 165 10, 18% Taos 12, # 
100 100 304 aor 
142. — Mixed 1. What are the two steps in dividing 
Numbers Divided. 16453 by 9? 
Written. 
2. 34768 +8 =a ST OLCSS. 
5 
8. Divide 73294 by 12 ee , 
7 ¥é 
a UA1Sp I = & a. }of 16453 = 182, 
5. How many times is 11 contained and 73 remaining. 
in 8764? b. Zot TZ = 4% of A= fe 


6. Divide 2893 by 26. To what common fractional unit are both 
dividend and divisor changed ? 


Process. 
26) 280% 7. 3672 + 24 9. 47232 + 105 
m4) 4 8. 8462 + 39 10. 69481 + 216 
104) 1159(11y4%; ll. If 75 boxes weigh 847,55 lb., what 
= will one weigh ? 
104 12. 89 rd. = 14683 ft.; 1 rd. = x ft. 


rbzg0f 15 = 104 


74 


143. — Dividing by 
Mixed Numbers. 


Process. 


$ 248) $ 1428 

oe 8 

195 ) 11424(58 
975 


FRACTIONS. 


Oral. —1. If one chair costs $248, how 
many can be bought for $1428, and how 
much will remain ? 


2. Why do we change dividend and divi- 
sor to 8ths? 3. The remainder is always 
a part of what? 4 What is the unit of 
the first dividend? Of the second? Of 
the remainder ? 

Written. —5. How many times is $428 
contained in $2500 ? 


6. 1 sq. rd. = 2721 sq. ft. ; 1728 sq. ft. =a sq. rds. 
7. Find the quotient and remainder; 2000 cu. ft. + 182 cu. ft. 
8. At $ 0.873 each, how many spoons can be bought for $75, and 


how much remains ? 


9. Of $525 I spend as much as I can for bicycles at $1257 each. 
With the remaining money, how far can I travel at 2 cents a mile? 


144. — Fundamental Processes applied to Fractions. 


PRACTICE TABLE. 


. b. c d. é. ag q. | h. ds 
1. 2 3 i} 23 63 152 516} $ 100 
2. 3 2 3 12 181 284 4932 250 
3. 2 3 a O35 975 213 6412 500 
4. i Ts a5 62 103, | 642 8274 576 
5. 4 ops Fire 98 152 853 9364 600 
6.| 3 35 BL 84 212 | 905% | 14641 640 
1.) 45 ee 13 dit If 164 | 2525; 720 
8} 44 pe Te 93% | 124 | 721 47693 800 
9. 2 3 zis | 104, | 183 362 84613 960 
10.| i4 + es 348 | 2014 | 253 | 755055 L000 


FOR PRACTICE. 75 


To tHe TEAcnER. — Each of the following 45 combinations may be applied to each number in 
the designated column so as to furnish ten examples, which may be assigned consecutively from 
I to 10, making 450 in all. How much to use depends on the degree of accuracy and _ facility 
attained or desired. 


Addition. Subtraction. Multiplication. Division. 
l. 6+2 ll, 20 —e Sher Se.0 ol. ba 
2. ¢+3 12. f— 54 Boe LG 32. a+c 
38. d+ 13. b—e 23. axg 83. b+¢ 
4, b+c¢ 14. d—corc—d 24 bofd 34. e+d 
5. c+d 16. f—e 20.16: % oh 35. f+e 
hep ¢-d 16. g—f 26. bx exd2 80. oF 
1 e+f 17. 947 —g ahs OO Gee ST 9 
8. f+g 18. h—g 28. fxg 88. h +75 
9 e+ftyg 19. g—d 29. 67 xh 39. i+e 
10. e+ft+gt+h 20. g—e-f 30. g xi 40. i+g 
Mixed. —41. b+c—id 42. g—(e+/f) 43. cof f+e 
44. gy+exf 45. (b of e)+(c of f) 
145.—To be 1. An heir gets ¢ of an estate, then loses 3 
Formulated. of his share. What part of the estate does 
Written. he keep ? 


2. I buy at 20% discount. What is the total cost to me of goods 
sold regularly for $1.42, $3.98, $57, $0.162, and 9 pieces at 
$ 0.314? 

3. If 84 tons of coal cost $487, what is the cost of 68 tons? 

4. Property which cost $5000 is rented for $434 a month; what 
is the annual income to the owner after paying a tax of $15 ona 
thousand ? 

5. Three cheeses, weighing respectively 344, 423, and 474 Ib., 
were sold for $20.60; what was the price per pound ? 

6. J. F. Sampson bought 721 bu. potatoes at 621% a bushel, and 
sold 3 at 641, the remainder at 75¢; what did he gain ? 


76 FRACTIONS. 


7. An electric launch was sold for $285, or 32 of the cost. Find 
2% of the cost. | 


8. 2 of a ton of hay (@ $20 pays for 14 tons of coal at how much 
a ton? 

9. 161 ft. of 2-in. pipe @ 61¥, and 1020 ft. of 1-n. pipe @ 41f, 
are exchanged for 120 lb. tubing at 1149, and 134 ft. @9¢. What 
is the difference in value ? 


10. Two trains start together in the same direction. How far 
apart will they be in an hour if one goes a mile in 1,3, min. and the 
other in 85 sec. ? 


146.— Fractional Parts Compare with 100:— 
of 100. 125055 25, 91.5, 920, ee LO oo 1 0, 
At sight. Oi. 1D) Ole GS 


Repeat rapidly, until thoroughly learned, the values of the following 
parts of 100: — 


Quit. bee oi euros ieee y fae ee is tee BAT a Fa OL 
Sas 3 Bp! 5 5 y SFU 12 16 20 2D ee O 100 
42 4 Hl es Oe oe on Pea a Sake Me ei oe i! 
ie Pa 5 «6 6 he eke 8 == ARH) 12 16 DRY Ue ANE yl 
5a he AS See 0 oe See ees a) ee 
ak} 10 12 16 20 PiDmeDIO ee 16 40 3 16 16 
Bris eZine 4207 ea eae 10s cia ey hs Seed Be 
ert sO 6 6 8 8 8 LD 16 12 40 16 L 2 


11. A nurseryman sells 2500 strawberry plants @ $6 a hundred. 
They cost him 3 as much to raise, and he gives an agent 4 of the 
profit. How much does he gain ? 


147.—To Find “Compare one number with the other in each 
what Part or what of the following columns. Thus: — 
per cent One Num- 8 is 3 or 834% of 24. 
ber is of Another. 24 is 3 times 8 or 300% of 8. 
The ratio of 8 to 24 is 4. 
The ratio of 24 to 8'is 3. 


Saye 


1 
2 
3 
4. 
5. 
6 
is 
8 
9 
10. 


148. — Finding 


NUMBERS COMPARED. 
[T. IS: 

25, 64 90, 18 
314, 10 Oe) ot, 
3,4 2 gal., 3 qt. 
72, 60 30, 50 
37k, 64 4, 0.16 
3, & 0.68, 0.51 
3 103, 5} 
10%, 20% = Foo Yor 
$0.75, $1.25 $1, 100¢ 
374, 874 334, 100 


Va 
Palen ie 
1 wk., 1d. 
5 min., 25 sec. 
144, 48 


16 


_80 
100° 100 


77 


177 2 a ir 
Ax 3 XS 


0.98, 0.31 
1, 200 

$1, $1.50 
1.831, 1.00 


What number compares with LOO as — 


what %. 

4 with 28 15 with 40 6. 85 with 105 = 3d.with 1 wk. 
2G) 280 6% 142 7. 4in.withiyd. 2mo.with1 yr. 
25 “ 150 24 “ 42 BEE OZ Stee Llores dai. 88s cL ini 
LE ois 8: aS eee one ue Looe t feta. LOU 1B." er 1 1” 
18 “ 54 PGuee meceeeme Or ite Ca al at fee a coe gl 

149.— A Part Given; 1. 12is tof 5. 28 is 2 of & 
the Whole Required. 2. 16is tof a 6. 36 is 3 of & 
Oral. 8. 24islLof a 7. 72 is ¢ of & 

4. 19isiofz 8. 100 is 19 of & 

9. What is 7 of 960 ? 11. 450 is 43 of what? 

10. What part of 23 is #? 12. 175 is 23 of what ? 

13. 47 or 85% of x tons of coal were sold. 

14. What part of 1200 is found in 480 ? 

15. 33 is + of what ? 18. 7 is $ of what ? 

16. 123 is 3 of what? 19. What part of 2 is 7; ? 
17. 142 is 2 of what ? 20. 183 is what part 62} ? 


78 FRACTIONS. - 
150. — To find the 1. 16 is 4 or 50% of x 
Whole when a Part or 2. 241s 3 or 75% of x 
Per Cent of it is known. 3. 32 18 2 or 662% of x 
4. 40 is 2 or 831% of x 
5. 56 is % or 874% of a 
6. 20 is 25% of w 9. 2118 10% of x 12. V25 is 5% of & 
151s 124% ofa 10. 321s 331% ofa 18. 361s 18% of x 
eeu is 3710 ofa Ill. fis 20% ofa 14. 9? is 27% of x 


a 


151. — For Frequent 1. 35 of 10 10 x 10+ 45 
Practice. 10 is = of what number ? 
Oral. 2. Change to fractions of a dollar in lowest 


terms: $ 0.124, $ 0.375, $ =25,, $ 0.662, 831%. 
3. Express in cents these parts 6. What is 1000 times — 


of a dollar: — $ 2 + + 7 
aoe ern Rae t. Give the fewest cents that 
will pay for 1 when the price of 
4. Of 1000 find — 12 is— 
+ 4 32 § fF 1p 16 18 20 25 


30 oo 38 40 42 
5. Use each of the following 4° 50) 65 70 88 


numbers as divisor of the one at 8. Add the following  frac- 
the right or left of it: — tions : — 

ou 1d Liesl 65 26 3 4 +9 4 rs 

& 10 621 1000 125 2000 2 4 335 4 oy 


Give rapidly the following parts of 100 
9. 4, 4, 4, etc, to. 10. 2, 3, 4, 5s $, etc., to 49 


2? 39 4 
152. — General Give numbers when needed to explain your 
Questions. answers. 


1. What are the processes of combining 
fractions ? Of separating them ? 
2. Of what use is it to change the form of a fraction and not its 
value? Explain the principle. 


PROBLEMS. 79 


3. State the method of finding the sum of two fractions if their 
units are not alike. 

4. What is meant by “higher terms” and “lower terms ” ? 

5. Show a connection between % and fractions. 

6. Why is the product of two fractions less than either? 7. How 
can a quotient be larger than its dividend ? 

8. Compare common and decimal fractions. 

9. In working with fractions what is the need of finding a greatest 
common divisor? <A least common multiple? 

10. What is the difference between a fraction and a fractional 

unit ? 


153.—FProblems for LHzxplain exactly how you get each result : — 
Analysis. 1. 34% of certain telegraph lines are 
under ground. What % are above ground? 

2. A foundry uses 100'T. of Swedish iron to 50'T. from other 
sources. What part or % of each class is used ? 

3. $160 was 54, or 16 % of the profits, which were a. 

4. 23 % of the stock was glassware, 69 % was china. The rest 
was in brass goods which were a % of the whole. 

5. The 5000000 sq. m. of the Arctic Ocean are what % of the 
area of the Pacific, which is 16 times as large ? 

6. I gain 100 % on 4 my goods and sell the rest at cost. How 
much do I gain on $100 invested ? 

7. In a 36-column newspaper what part of the whole space would 
be filled by 20 columns of advertisements ? What per cent ? 

8. I lose half that I have, and 25% of the rest. What I keep is 
what part of what I lose ? 

9. The board of a horse is $20, shoeing $1.25, harness repairs 
$0.25, use of carriage $3.00, new whip $0.50. Each item is what 
part of the whole? Give per cents when you can. 

10. ®P=2 %&% of 9? 23 —=2% of 8? 
§of 4% =1hofaG 10 % of $10 less 1 % of itself = x 


80) FRACTIONS. 


154. — For Oral 1, 10 =12 of what number? 10= 12 of 
Analysis. what number ? 


2. Give 4 and ;4, their least common de- 
GERD 
3. 2 of a hill is dug away. How many times as much remains ? 
3x#+tioft=a 1+2#=y. 
iL 


5. A train goes a mile in 12 min. How far will it go in an 
hour? .6. At 1m. in 90 sec., hae much in an hour? 7. 50m. an 
hour = how much a minute ? 

8. Express in lowest terms $9. 9. At the rate of 48 m. an hour, 
how long does it take to go 1 m. ? 

10. A mile in 124 min. is the same as 60 m. in .? 


155. — At Sight. 1. Reduce to lowest terms 34°; 5497. 
2. Compare results: 224.7 yards+7; 
224.5 yd. +7 yd. 
3. of LO=a% Sof 10=y. 4 4 league = what part of 3 
league ? 
5. é A. = what part of 12 A.? Of14 A.? 
6. 21s contained in 12 how many times ? 
7. Cancel mentally: $ of 8 x 14 of 12 = a. 
8. By getting a discount of 21 pay only $3.33. What is the 
regular price ? 
9. £62 + dey = 2. 
10. What is the least that will pay for 1 article, when the price 
per dozen is — 
$1.05 $110 $1.15 $1. 25 $1.30 $135 $140 $1.50 
Fae $2.00 $2.25 $250 $2.75 $3.00 $5.00 
1. 3rd. @ $1.25. 13. 31 in 25 & times. 
a 57 lb. @ $ 1:28, 14. 40 — 163 — 74 =4@, 


EE ee 


EXERCISES. 81 


156.— For Dictation. 1. Add 14, 21,° 58, 834. 
Results orally. a 10 43 4. 3. $ 13 — $ 0.625. 
4, 4121 + 874+ 6214 874411214121; 662 — 162 — 81. 
5. In a Fahrenheit thermometer what is the temperature when 


the top of the mercury column is ;°, of the distance in degrees from 
zero to the freezing point. 


6. After gaining #5, or 10%, I have $99. What had I at first ? 
7. Find 24 x 90. What is ;1, of it? 6% of it? 
8. What part of 100 is 97? Give 72, of 100; 33-; 545 74. 
9. How much for a dozen at 16 for a quarter? At 4 for 5¢? 
At 20 for a quarter? At3for10¢? At3 for 5¢? 
10. A newspaper weighing 4 oz. may be mailed for 1%; what will 
it cost to send — 
1 lb. 10 oz. 5 OZ. 41 oz. 8 oz. 
The rate for other printed matter and for seeds is 1 for every 
2 oz. Give the cost as before. 


157. — Fractional + 1. Compare the cost of — 
Measures. 7 A. at 10¢' a foot and 560 A. at 25¢ a 
rite: square rod. 


2. Corn is worth 531¢ a bushel. ae 

buys a bushels, with y cents remaining. If aid at a profit of 54, 
what will be the gain? 

3. Copper sells for £39 5s.aton. A pound is worth about « 
cents. 

4. At 34 leagues an hour, how long will it take to go 500 m. ? 
How far would a ship sail in a week ? 

5. Find the average of these prices for oats: 29%, 298, 283, 297 
d04, B03. 

6. If pork costs 12,35,¢ a pound, what is the profit on 4 bbl. 
retailed at 14¢ (200 lb. to a barrel) ? 


89 FRACTIONS. 


7 A barrel of 42 gal. will fill how many cans containing 17 pt. ? 


8. When $80 are earned in a month and 5, of it spent, the sav- 
ings of 2,1, years at that rate would be $2. 


th el ie contains 714 lb. tea @ 55%, 134 Ib. @ 35¥, and 93 Ib. 
(@ 27¢. The mixture is worth af a pound, and $10 would ani 
y lb., with z ct. remaining. 

10. 43 of a cargo is the captain’s share. It consists in part of 
14 cwt. coffee @ 28¢, 230 lb. cloves at 18¢, 180 lb. ginger at 103¢, 
15,000 gal. molasses @ 251f, 2580 bags sugar @ $3.121. What is 
the value of the captain’s share of these goods ? 


oa 


158.— Oral Problems. 1. Mention 3 fractions of a dollar which 
together equal $2; $0.50. 


2. What sum becomes $ 2 after spending 4, 4, and 8, of it? 


3. Which is more, 55, of a share or 5% of one? When the differ- 
ence is $10, what is “ite whole share ? 


4. Of the 8000 bills presented in the first session of the 53d 
Congress only 1500 were considered. What fraction was that ? 


5. A ship was insured for 50% of its value. #2 of the insurance 
was $4500. The ship was worth 2. 


6. I pay for a purchase with a $5 bill, and receive in change 
one of each of the current coins of the United States amounting to a. 


7. The back of a bench is 4, as high as the seat, which is 2 ft. 
Find the difference in inches. 


8. One plant urn holds 12 pk. How many such can be filled 
from 3 bu. of earth? Out of 34 bu. what part could not be used ? 


9. How much velvet at $4.371 a yard can be bought for the 
price of 43 yd. satin at $ 2.662 a yard ? 


10. < of a telephone pole is the part EA ground. 8, of this 
part is painted brown, and the remaining 7 ft. are painted white. 
What fraction of the whole is white? How long is the pole? 


QUESTIONS AND PROBLEMS. R38 


159.— For Rapid 1. Paid at different times 4, 4, and 54 of a 
Figuring. debt. The balance was $1170.61. Ww hat 
was the whole debt? 

2. 4 of an estate is divided equally among 14 persons, another 
4 among 9 persons. One of each of these shares is to be given to 
the heir of the remaining third. What part of the whole does he 
receive ? 

3. From a life-saving station to the end of the northern beat is 
235m. How many full eR will a surfman take in going and 

Eerening: if his steps average 2,2, ft. each ? 

4. The Minot’s Ledge light revolves twice a minute. It is hghted 
from sunset to sunrise. How many revolutions does it make between 
5.02 P.M. and 5.53 A.M. ? 

5. Four equal farms, all adjoining, are offered for house lots. 
Parts of each are sold as follows: 8, 58, 3%, 7%. Add the fractions, 
and tell what the sum shows. 


6. What part of a mile is covered by 22 revolutions of a wheel 
18 ft. round ? 


7. Divide 231 cu. in. into 3 equal integral parts; into 33. How 
else can it be exactly divided ? 

8. Just when does + of a common year end? 4 of a leap year? 

9. A bushel of potatoes is commonly 60 lb. A thousand 56-lb. 
bushels are what part by weight of 1000 60-Ib. bushels ? 


10. What other fractional units besides 4 and 4 express integral 
parts of 45,560 sq. ft. ? 


160. — Business Find the amount due on the following pur- 
Transactions. chases ; — 
Written. 1. 3) yd. silk @ $1.874; 84 doz. buttons 


[See pp. 8-9. ] (a, 154; 7 71 sticks braid @ 75¢; 23 yd. ribbon 
(a, 162. 
a 6470 ft. fencing @ 9,5,¢; 3 lots land, 10280, 7595, 8122 sq. ft., 
51g; 3400 bricks (@ ne 15 per M. 


84 FRACTIONS. 


3. Find the total weight in pounds : — 

1 'T. 33 cwt. 2560 02. 1 T. 5 lb. 8 oz. 5 cwt: 9 lb. 

4. How many pounds altogether will fill 600 4-o0z. bottles, 
360 2-0z. bottles, and 36 doz. boxes, each containing 20 2-gr. pills ? 
What fraction of this number would represent the same weight 
in avoirdupois pounds ? 

5-7. A. & H., B. C. & Co., F. 8. T. are the initials of three firms. 
Make out three bills from these entries on the day-book of James 
Graeme, who sold the goods. Use the present date. (See p. 40.) 


B. C. & Co., 231 M. ft. pine @ $55 
F. 8. T., 371 locust posts @ 624 
1090 split rails @ 183¢ 
A. & H., 43 M. brick @ $8 
1830 Welsh slate @ 101¢ 
A. & H., 864 ft. 2-in. plank @ 3¢ 
} cd. 5 ft. slabs @ $3.25 
F.S8. T., 500 ft. sheathing @ §$ 42.50 
11M. ft. flooring @ $20 
B. C. & Co., 22 bbl. lime @ 673¢ 
13 T. coal @ $4.75 
teaming 13 x 371¢ 
B. C. & Co., 73 M. ft. spruce @ $31. 
7 7 bbl. Portland cement (@ $2.75 
F. 8. T., 350 ft. drain pipe @ 624 
A. & H., 3 cd. hard wood (@ $8.75 


8. Rule paper for an account that you keep with John Holmes. 
It will show that he is debtor for all that is sold to .. by ~, and 
creditor for all that is paid to _. by ., as below. 


Joun HouMeEs Dr. Or 


189- | | | | 


Nay if Jo 8 aharew MAU atock 3/2\00 }\ 
Ly 4 River palwre (3 A. ) 


EXERCISES. 85 


Make out the account as above from the following memoranda, 
supplying different dates : — 

34 days repairing Holmes’s fence (@ $2.50. Credit him for use of 
his oxen same time @ $2. Sold him 23 bbl. apples at $1.40. 
Bought 3 hogs (732 Ib.) at 114, and 14 loads hay $15 (both of 
Holmes). Sold him 15 young maples at $1.064, and 3 hoops con- 
taining 35 ft. strap iron at 2 39, Holnes paid an $100. 

9, Draw off in proper form (p.18) a cash account for July con- 
taining the following items (cash must be credited with all that 
you take from it to spend): Left over, saved, or “on hand” from 
last month, $2.47, and had a present a ‘five dollars vacation money 
on the Fourth, besides the 75¢ I earned at the fair the day after. 
My ticket to Newburn was $ 1.38, the luncheon 30¢, and the express 
25¢: this was on the 6th. In just a week I had to spend all the 
rest of my money except 44¢; three dollars being for a hat and new 
ribbon, and all but the 44¢ for a ticket to Groveton. There I found 
waiting a money order for $15. 

10, Rule paper as on p. 18, and make up a balanced cash account. 
(“ Cash” is debtor for all that you entrust to it.) (Aug. Ist) Bought 
9 shares A. and F, stock @ $1093. Sold 8 shares @ $1093. 
(2d) Paid note of E. F. James, $128, with 3 yr. interest (8 x $7.68). 
(5th) Sold 8 bbl. lime @ $0.831. (9th) Sawing wood, 181 ed. 
(@ $1.20, paid to Tom. (12th) From Edwards, $56 for 54 squares 
roofing @ #. (20th) Taxes on $14,000 @ $14.25, paid. 27th) Re- 
ceived cash for 504 gal. oil @ 81¥. (29th) Sold 2 cases eggs, 48 doz., 
@ 162¢. 


161. — Fractional Figure the exact value of — 
Measures. 1, 290 ft. hemlock boards @ 13¥. 
Written. 2. 19 ream @ 2 a sheet. 
[See pp. 8-9.] ny 3, 25 bu. @ 2 qt. for $1. 


4, bas for 1 yr. (even weeks) @ $ 2.374, only Sundays excepted. 
5, v g. gr. hooks and eyes at 5% a doz. 
6. 4% of a 235-lb. barrel of sugar @ 4,5 


86 FRACTIONS. 


7, 112 long T. @ $5 ashort T. 8 24 oz. opals @ $3 a pwt. 
9, 64 lb. avoirdupois @ 10 gr. for $0.01. 

10. A pound bar of silver, less 760 gr., @ $7.80 a pound troy. 

11, A bottle contains 20.79 cu. in. of varnish. What is it worth 
(@ 162% a pint? 

12, If the duty were 4 the cost of certain fabrics, the duty on 
enough to cover 4 a square rod @ $2 a yard would be x dollars. 

13. Find the cost of 164 sq. yd. @ $0.375. The value of 1 sq. 
it. = Ww. 

14, If roofing that would sell for 23¢ by the sq. in. can be had for 
+ as much when bought by the sq. yd., what would be the cost of 
1 square (100 sq. ft.) at the yard rate ? 

15. How many 30 ft. rails will reach a mile? 2 of such a rail is 
how many yards long ? 

16. 90° of latitude =a miles. The distance round the world on 
the same circle would be y mules. 

Deny Gi Lt O02 glee ee noe 

18, 3, A. =«sq. rd. =ysq. yd. How many squares 6 in. on a 
side would cover 1 sq. ft.? 1sq. yd? 3 A.? 

19. If a square top desk is 222 in. on a side, what is its perimeter ? 
4 of this distance from one corner would reach how far beyond the 
second corner ? 

20. A picture moulding is 14 i. wide. How much farther is it 
round the outside than round the inside of the frame ? 


162.—Denominate 1, 2,1, lb. troy sold by the 3 at 30% = 2. 
Numbers. 2, 154, tons cream of tartar fills how many 
Written. — 4-0z. boxes ? 
3, 323 yd. lace billed at $87 costs how 
much a yard ? 
4. £10 4s. is the amount of an invoice of a case of cutlery con- 
taining 12 boxes (@ w shillings a box. 


QUESTIONS AND PROBLEMS. 87 


5. 17s. 1s what part of a pound? of $4.866? 17 of $4.866 = 
how many shillings? How many pence ? 

6. Show in figures whether a nautical mile is more or less than 
1} times a common mile. 


7. One piece of cloth is twice as long as another, which is reduced 
to 1,5; yd. by cutting off 93, 31, and 23 yd. How long is the first 
piece in yards and inches ? 

8. With hay at $13? a ton, what fraction of a ton is worth 


$13? How many pounds ? 


9. When 115 votes are in favor of a project and 46 are against It, 
what part of the whole are opposed ? 


10. Find the profit on 10 bales of wool bought at 164¢ and sold 
at 163¢, total weight 224 cwt. 


163.— Ast, Written 1. At an auction, one buyer bids ;% of the 
Results; cost; another, 4+. The difference is $75. 

2d, Oral What is the: lower bid? 
Analysis. 2. From a grove of 40 trees 133 bu. chest- 
nuts are saved for market. Supposing squir- 
rels to have taken 3 qt. from each tree, what fraction of a bushel did 


each tree bear? (Make statement.) 

3. 575 ft. of fence are to be built. The posts are 1 ft. through, 
and the space between posts is 3} yd. How many posts will be 
required ? 

4. How long will it take to perform a task of which only 7 can 
be done in 3 d.? 

5. Mary can do the washing in 3 h.; Sarah,in4 h. What part 
can each do in an nour? If they work together, how long should it 
take to do it? 

6. John hoes a corn-field in 4 d. Last year his brother could 
work 4 as fast; this year 2 as fast as John. How much can each 
do ina day? How long will it take them together to hoe the field ? 


88 FRACIIONS. 


7. Four men are partners in business. Three ge $ 21,000 
capital; the other gives only his services, and has ,% of the profits. 
Two furnish ? and 3 of the capital; how much are does the 
other furnish ? 

8. If a glass jar contains a hundred thousand fish-eggs, how 
many jars will hold 8,000,000? If 42 are hatched, how many on an 
average are lost from each jar ? 


9, The catch of shad for a certain period is valued at $ 145,000. 
What part of this is $4000, the cost of hatching the eggs and stock- 
ing the waters ? 

10. When oysters yield 11 gal. to the bushel, a 25-gal. barrel can 
be filled from a bushels in the shell. 


164.— To Illustrate 1, A square floor contains 64 sq. yd. Draw 
by Diagrams. it, and mark it off in squares. Show the 
dimensions of a square carpet that covers ;%- 

of it. 


2. Represent a street having a branch 32 as long. 
3, Show how many sq. ft. ini sq. yd. + 2 sq. yd. + 12 sq. yd. 
4, 1 rod=16} ft. How far is it round 3 sides of a rod square ? 


5. eee the square inches in a square foot. Draw a heavy line 
of =%, of them. 


been | 


ae 
oa ee ebay 


x Sq. ft. contains how much more than 144 sq. in. ? 


ff ‘ai distance between two fence posts is 10 ft. How many 
34 in. slats, 5 in. apart, will be needed ? 


8. Set the hands of a clock dial at 10h. after 411 h. 


9. A postage stamp covers 13 sq. in. How many times as large 


is an envelope of 21 sq. in. area ? 


10. The circumference of a wheel is 12,57, in., which is about 31 
times the diameter. Draw the diameter. 


QUESTIONS AND PROBLEMS. 89 


165.—For Written 1. 37h yd. @ $32 a yard. 
Work. 2. Clittord can mow the lawn in 1 hour. 
From dictation Clifford and Leonard can do it in 24 min. 
if possible. In 1 min. what part is done by Clifford? By 
Chitford and Leonard? By Leonard? How 
long would it take Leonard to do the whole ? 

3, Steaming from New York harbor at 10 a.m. Saturday, and 
reaching Holyhead in 54 min. over 51d., what time is it in New 
York? 4. If morning comes 42 h. sooner in Holyhead than in New 
York, what time will it be by a Holyhead clock ? 

5. A piece of ground sells for $12,600, which is 3 of what it 
might bring if properly drained. How much would be added to 
its value by draining ? 

6. A tent contains 40 metres of canvas } metre wide. If the 
stuff were 2 metre wide, how much would be required ? 


1. 72 i suk # yd. wide = 6,8, yd. of what width ? 

8, 2 of a box - pens costs ond ; 18 boxes cost what ? 

9, A man who received 7 of his father’s property gives to his 
own son ? of what he cee Who then has 3% of the whole ? 


10. Two men require 8} days to take account of a stock of goods. 
Six men would need what time ? 


166.—For Written Make statement and cancel when possible : — 
Work. 1. How many tables can be made in 15 wk. 
at the rate of one in 52 d.? 
2, 239- 21—=a 38 ex=1 xt of 142 multiplied by 233. 
4, What mixed number is the Sapaent of 92742 + 1155? 
5 


. What fraction is the quotient of 290 + 2.%%, ? 


6. A contractor is to receive in three equal payments $ of his 
expenses in digging a cellar. What may the first payment be after 
6 men with horses have worked 194 days at $2.25 aday? 17% The 
wall cost $872 to build, and the evading $ 286.55. On the same 
conditions each payment would be what ? 


90 ' DECIMALS. 


8. In sowing a field, one kind of seed is used at the rate of he 
bu. to 5 acres. What will be required to sow 222 acres, using # as 
much to the acre as before ? , 

9. What will be the profit on 1200 rolls of wall paper costing 
15¢ a roll and selling for 14 as much, — excepting + of the lot which, 
being damaged, sells at cost ? 

10. A sold goods at a profit of 1, to B, who made a profit of 2 
B sold them to C for $738, which aa 11 times what they cost fee 
What was the cost to A? 


Decimal Fractions. 
167.— The Orders; [Review §§ 4-15.] 1. By a decimal Petar 


the Point. we mean —~ 
2. Gomes in value any one digit with 
another written at the left of it. . 


3. Compare the values of the 3’s in 33.; in 3.38; in 0.33; in 0.033. 
4. Explain which numbers are integers; decimals; mixed decimals. 
5. The value of a figure depends on what two things ? 


7654321.234567. 
6. What figure stands for tenths ? Hundredths ? Thousandths ? 
7. Compare the position and the value of the 3’s. Of the 5’s. 
8. Of what orders are the 6’s ? The 7’s? 
9. What is the use of the decimal point? 10. Why should the 
2d figure at the left denote tens, but the 2d figure at the right, 
hundredths ? 


168.— The Denom- 1. The denominator of a decimal fraction 


inator. is .. 2. Why need we not write it ? 
oe ae 
3. Are 58, 7$o, z°77 Common or decimal 
fractions ? 


4. How would you write them decimally ? 

9. Compare the number of places each takes up with the number 
of ciphers in its denominator. 

6. Make a rule for pointing, so as to show what the denominator is. 


COMPLEX. 91 


169. — Reading 1. What part of 140.040 should be read 
Decimals. first ? 
2. Where is “and” used in the reading ? 
3. Where does the numerator begin, and where end? 4. Which is 
the easier way of finding the denominator of 0.040, — 
(a) By counting from the point —“tenths, hundredths, thouw- 
sandths” ; or (6) By imagining a 1 with three ciphers annexed ? 


. 


Read : — 

6. 0.307 0.0307 0.010 6. 0.710 64.0700 500.005 

7. 330.03 . 3005.075 8. 37290.760009 808.08 

9. 94.07508 6.06008 10. 64.000109 0.009004 

11. 0.407501 900.09 12. 6000.006 0.06006 

170. — Complex Since 84 per cent, or 84%, means 84 hun- 

Decimals. dredths or 0.84, in what two ways might we 
read — 
1. 0.06 334% 0.162 2. 0.87% 624% 0.034 


3. 34 thousandths is written 0.003}. To which order of units 
does the + belong? 4 Give other complex decimals. 


171. — Writing Oral. —1. When there is no integral part, 
Decimals. what may be written in its place before the 
decimal point? For what purpose ? 
2. If the denominator contains three ciphers, how far from the 
decimal point must the numerator end? 38. If that numerator con- 
tains but one figure, in what place must it be written ? 


Write from dictation or at sight : — 


4. § thousandths 9. 804 hundred-thousandths 

5. 17 tenths 10. 400 and 4 ten-thousandths 

6. Three and a fifth % 11. Forty thousand forty millionths 

7. 3075 millionths 12. Seven hundred six thousandths 

8. 4a hundredth 13. Two million 71 and 404 millionths 


92 DECIMALS. 


172. — Problems. Try to show that — 
1, A millionth of a mile = about 1, of an inch. 
is ; 
2. ——___ 
1,000,000 
38. A millionth of an acre (43,560 x 144 sq. in.)=a square about 
21 in. on a side. 


yr. = about }. a minute. 


4. A hundred-millionth of the distance to the sun =WW. 


5. What is the name given to 0.0000001 of the distance from the 
equator to the pole ? 


173.— Reducing to 1. Change 12 to smallest terms. What 


is 
Smaller principle applies? 2. Prove that 58%, =. 
Terms. 3. Compare the numerators, the denomina- 


tors, and the value of 0.90 and 0.9. 

4. Dividing both terms of a fraction by 100 has what effect on its 
value? 5. Omit the zeros at the right in 0.360 and 0.400. How is 
the numerator affected ? The denominator? ‘The value? 6. State 
the principle. 


Read as printed; then in smallest decimal terms, telling what g.c.d. 
You Use: — 


7. 0.0400 6.6450 0.03070 9. 84.700 6.040000 

8. 7.09000 0.060 0.5000 10. 921.7600 800.80000 

174.— To Larger 1. Explain any change of value made by 
Terms. annexing a cipher to 8; to 120; to 0.3. 


2. Explain why 3, = 53° and 0.3 = 0.30. 
38. When both terms of a fraction have a zero annexed, how is the 
size of the unit changed? The number of units ? 


4. How may you know the number of zeros in the denominator of 
a decimal ? | | 

5. Js the value of a decimal affected by annexing zeros, as in 0.4 
changed to 0.400? Explain the principle. 


REDUCTION. 93 


Read as printed ; 


GO U.0 2:20" O07 0,101 


then with one, two, and three ciphers annexed ; — 


7. 37.4 0.008 15.0 245.6 


8. Read the preceding numbers, and announce each change of 


value when the point is moved one place to the right. 


left. 
175.— Decimals Simple. 
to Common 75 3 
: 0.75 —_-err- = 
Fractions. 100 4 


1. Explain the first change; 
then the second change. 


2. Give directions for the two 
steps of this process. 

Change to common fractions. 
Give steps orally. 

o 0:2, 04° 05 0.67 0.80 

4. 0.25 012 0.50 0.24 0.70 

5. 0.08 0.125 0.45 0.480 

6. 0.875 0.00875 0.0064 

(Use small factors in reducing 
to lowest terms.) 

7. 0.000125 0.0625 

8. 0.01728 0.0675 


0.375 
0.4375 


177. — Written or 
Oral. 
0.412 


9. To the 

176. Complex. 
ay RUS et Saar ae eed 
100° 3645400) 16 
1. A. complex decimal is WwW. 
2. 183% means .. 38 Multi- 
plying both terms of - i by 4 


has what effect on the value? 
Why? 4. On the form? 5. State 
the principle. 6. Explain the 
whole process. 


Change hundredths or per cents 
to common fractions :—. 
T. 383% 0.162 662% 
8. 0.831 0.081 50%481% 
9.121% 0.374 871% 
10. 0.03) 41% 0.064 
ll. 62% 25%+4+61% 0.314 
12. Subtract each of the pre- 
ceding from 100% 


Change to common fractions :— 
1-10, 433%, 
0.933 


11-20. Subtract each of the preceding from 100%, and 


0.56} 
0.912 


621% 
225% 


0.683 
0.291 


814% 


change 


the remainder to a common fraction, 


94 


178. — Common 
Fractions to 
Decimals. 


Written. 


* = what decimal ? 


Method. 


8) 7.000 


0.875 


179. — Reducing 
to Complex 
Decimals. 


Oral and Written. 


Method. 
a. bd. 
12)5.00_ 12)5.000 
0.412 0.4162 


DECIMALS. 


ae ce Roms 700 700 
1, W rite decimally : i9 oo 7800, 
2. What is the difference 


7.00, and 7.000 ? 
or 4 of 7000 thousandths = @ thousandths. 
Write as 100ths, LO0O0ths, or 10000ths, ete. 


ne 6. 48 8, 8, 10, 18 
5, ff 


1 
20 2 
13 2 
40 1. 8 


0 
0 
between 7., 7.0 


7 
0 9. a5 


1, Give the prime factors of 100; 1000. 
2. If the divisor has a prime factor not in 
the dividend, how is the quotient affected ? 


3. Why is not 12 an exact divisor of 
5 x 100 or 5 x 1000? 
4, Change 5, to a decimal. 
5. Compare the value of the 


C. quotients in @ and b. Are 
12)5.0000 both exact ? 


0.4166 + 6. Is the quotient in cexact ? 


The + stands for what ? 


Change to complex decimals of 


three places : — 


Change to incomplete decimals 
of four places : — 


4 3 ay 1 by 6 yall 

Tet 18-22 2 #3 2 & H 
iby fs 1 3 tbe 

aoe 23-27. ve sty we vs 4 


180. — Rapid Reduction. 


. 18 betel aes 
Oral. Otgs . are 
Change to hundredths or per tery oo eh ee 
eae Bay Bay Oly sh se ly a 
cents . oy vo $ te Ve 

Lie oe a ee Gj. 25s BOTS SiGe han 2s 
PGi MP ALS G20! FE Ve cio 
1 Le oe LO el. _3 Wee ee Se Ses Oe ed 
2. I § 56) Th 25 7. Oy 16) Tl “oe 


ADDED AND SUBTRACTED. 95 


Change to the other form :— 

8. 12 0.814 47 912% H 
10.14 561% UW 0.314 581% 
ll. Give the difference between the last fifteen numbers and 100%. 


13 23 ml ies 2 
16 ABE% 16 413 


tpl 


DrecimMALs ADDED AND SUBTRACTED. 
181.— For Addition. Without copying, write the sum of each 
Written. column, and of each line. 

1. 2. 3. 4. 5. 

6. 96.475 + 186.32 + 0.4875 + 0.64985 + 396.47 

7. 83.8 + 62.38794+2.938 +3.207 + 82.379 

8 542 + 48 +8479 +0.0459 + 64 

9. 16.785 + 9.54 +653 +9.642 + 180.09 

10. 4.09 + 72.683 + 2.946 + 8.78314+ 34.769 


182. — Written. Rewrite in columns, as integers and deci- 
mals; then add. 
1. 16.372 83 79.42 141 862 21.054 
2. 2164 B42 19.3794 12), 97335 16745 
3. 25525, 0.847 964, 2912 5 621% 
4, 3 874% 0.758 934% a 121% 
5. 0.3 2.08 +i poe 116% 6.4837 
183. — Oral. 1. How may common and decimal fractions 
be added ? 


2. Compare the denominators of the amounts in the last two 
exercises with the denominators of the addends. 

3. What steps are to be taken in finding the sum or the difference 
of complex decimals ? 

4. Define the two ways of combining numbers. 

5. When part of a number is given (0.57 is part of 0.96), how 
is the other part found ? 


96 DECIMALS. 


184. — For Find the difference, — first explaining whether 
Subtraction. the denominators must be alike : — 
Written. 
if: 2. 3. 4, 4 6. 
7. 3.64 —1.873 9. 41. —13.074 ll. 6.389 — 0.497 
8. 1.9 — 0.3694 10. 987— 4.8 12. 2.641 — 0.0994. 


18. Take seventeen hundred eight ten thousandths from twenty- 
four and six thousandths. 


14. From eighty-six tenths take forty-three thousandths. 


185. — Written. 186. — Written. 
Find the difference - — Carry results to three places : — 


. 17.88 300 6. 94 7968 
SAGA 1 ae Sie CAG, 
8 


1 . (64 + 2.35) — 58 
2 

BigP 87h ese mees. 

4 

5 


(0.9 + yoo) — Gbia + 9.72) 
. (100 — 374%) + (0.625 — 4) 
[1.00 — (0.08 + #s)] —4 

. [44 —0.912) + 22] — 0.162 


#1021 0.851 coe Os el ae so 
. 0.1 0.0831 10. 64 § 


on 
S) 
nk oN 


DEcIMALS MULTIPLIED. 


187,— At Sight. 1. Compare the product and the multipli- 
cand when the multiplier is 1. When it is 
more than 1. When less than 1. 

2. Muitipheation is WV. 

3. Moving the decimal point one place to the left has what effect 
in 18.4? In 015? 4 What is the shortest way of dividing 247. by 
100? Of finding 0.01 of it? 

0. Read quotients after dividing by 10. 6. By 100. 

72.46 18. 0.875 625. 3.7 0.0334 9. 
7. shot 2.462 8. ssa Of 37.6? 9. 0.0001 of 3500. ? 
of 327 ross of 0.9? 0.001 of 25. ? 


MULTIPLIED. 97 


188. — The Process 
Analyzed. 


.2. To find 0.7 of 50 
JOC ae Ee 
aeOr 00 "6 ie = 2 
or 
BS DAD ans 
bot tT -« 50 0.1 of 360 =a 


4. To find 0.9 of 0.03 
to Of tia = robo = 9.027 
Compare the denominators 
and account for the number of 
ciphers in each. 
5. Count the decimal places in 
the factors. Why are there three 
in the product ? 


1. Having found ;4 and 0.001 by moving 
the point, how would you find 58 or 0.015 ? 


To find 0.15 of $300 
zi Of $300 = 
phy Of $3800=_.. x 15= Fe 
or 
$ 300 x 15 = 
$ 300 x 2és = qin Of ~ = Fa 


The product contains 
as many decimal places as 


its factors. 


6. How many decimal places in the product of 0.46 x 0.388? of 
0.372 x 0.471? Explain with common fractions. 


7. A product has 7 decimal places; one factor 2; the other 2. 


189.— For Rapid 


Analysis. 
1. 0.06 of 200 5% of 500 
2. $90 x 0.9 12% of 1000 
3. 0.6 of 0.6 0.12 of 0.12 
4. 0.08 x0.5 20% of 60 yr. 


Show what process you use, and see that your 
result is a reasonable one. 


6. 2.5.0.6 
6. 80% of $400 16% of 40 
7. 331% of 360 
Goin set 2 


i Ua 


0.2 <x 0.2.0.2 
200 x 0.008 


9. Had you first to multiply or to divide in these examples ? 


10. A man of 50, spending 30% of his life abroad, is at home a yr. 
11. 2% of $5000 being counterfeit, $a is good. 


12. 23 x 375 = 8625. 
2.3 xX 3875=2 
w x 3.75 = 0.8625 


LGa. 2 9 Ot = Y 
2.3 x 3.75 =2 
230. x u = 8625. 


14. How would you write the product of 0.02 x 0.004 ? 


98 DECIMALS. 


190.—For Written 3. 3.468 x 2.008 7. 0.36 x 3.6 x 36 

Work, 4. 0.81 x 5.076 8 $800 x 2.4, less 75% 
1. 8.47 x 9.432 5. 0.0371 x 12.50 9. 8% of 0.08 x 8000 
2. 0.84 of $9.60 6. 1.8%8x 0.360 10. 14m.4 0.64 m. 


191.— Problems to 1. 218 1b. sugar @ $0.061. Deduct 1%. 
Dictate. 2. A gross of blank books at $ 0.112 each. 
Written results. 3. A man’s salary is $1200. If reduced 
121%, it becomes 2. 
4. Some coal is 84% slate. The pure coal in 1500 T. is & tons. 
5. 120 yd. cotton @ $0.07125. 6. 306 millionths x 17 ten- 
millionths. 
7. 0.375 T. hay weighs x lb. 8. The square of 0.75 is what ? 
9. An $8000 house rents for 12%. The monthly rent is 2. 


10. The owner has this house insured for 3 of its value, or x, pay- 
ing 2% or y to the people who insure it. 


DrctmMALs DIVIDED. 


192.— The Process [Review pp. 70-71.] 1. One of the 8 equal 
Analyzed. parts of $0.48 is a. 
2. 0.06 is contained in 0.48 & times. 

8. What decimal part of 6 is found in 4? 

4. Whatis division? Divi- 
sion of fractions ? 

5. Explain division of deci- 
mals by moving the point in 
17.28 to right or to left. 
6. With what divisors can 
you find the quotient in that 
way ? 


Multiplying or dividing both divi- 
dend and divisor by the same num- 
ber makes no change in the quotient. 


DIVIDED. 99 


To divide 0.144 by 0.09 7. In the example at the left 
Process. how is the divisor changed from 
0.09)0.144 = 9.)14.4 0.09 to 9.? For what purpose ? 
1.6 8. How and why must the divi- 
To divide 63.44 by 25.6 dend also be changed ? State the 
ENS principle. 9. After dividing 14 
2.474 by 9, how many tenths in all 

25.6) 63.44 = 256. )634.4 remain to be divided ? 
a5 10. Explain the process shown 
102.4 in the second example. When 
20.00 there is a remainder, how do you 

17.92 continue the division ? 

2.08 


11, How may you always have an integral divisor ? 


12. Give directions for division of decimals in five steps : — 
I. Setting down. II. Pointing. III. Dividing. IV. Placing 
and pointing the quotient. V. Managing a remainder. 


193. — Written Notice whether the quotient will be larger or 
Exercise. smaller than the dividend. 

1, 21.6 + 0.006 6, 102.01 + 1.01 + 12.5% of 100 

2, 0.4913 + 1.7 7. $8.281 + 9), — 6.25% of $ 3.00 
3, 2.1952 + 0.028 8, 4.096 + 0.0064 + 0.82569 + 28.7 
4, 1.521 + 3.9 9, 67.24 x 82% — 67.24% + 82 

5. 0.6345 + 0.009 10. 400 + 0.662 + 876.16 + 0.296 
194.— At Sight. 1, 38.025 is to be divided into 195 parts. 


Will each part be more or less than 4? 
2. How many places would there be in the quotient if the division 
were exact ? 
3. How would you write decimally the quotient of 3 in 100? 
4, The quotient of 6 in 0.1? 
5. Add these parts of 100: 183%, 6.25%, 0.624, 54%. 


100 DECIMALS. 


1% of itis... #5 0f1% of itis... 14% is — 

1.5% of $10 =a. 0.5% =_.* 4% of $12 = 
1—0.9=a. Find the difference between 0.009 and 0.01 
0.1—001=a. 10. 0.01 — 0.002 = y. 


SO Soe 


195.— For Dictation. 1. A camel goes 3.5 m.an hour. How far 

in 10 h.? 2 Express 18 h. in hundredths 

of a day. 3 Express 200 min. decimally in hours. 4 Is a week 

more or less than 0.02 yr.? 5 Reduce to lowest terms 0.00004 

6. In a month there are about 24 million seconds. How many in + 
of a month ? 

7. The interest of $1 for 1d. is 4 of a mill. How would you 
write it decimally ? Did you choose a complex or an incomplete 
decimal? 8, Find the interest of $1 for 1 mo. at the same rate. 
9, How would you write decimally $itig +5d: 10. 1 mill is 
what fraction of 10 eagles ? 


196. — For Analysis. 1, £0.11 + £2=~2 shillings. 
[See tables, p. 8.] 2 V4—0.12 =o. 
gn Lies. 
621 1000 
4, 12 pence (d.) make a shil- 8, 10% of a ream = x sheets. 
ling (s.). Which is more, 1d. or 9. A quire= 27% of a ream. 
£ 0.004 ? 10. What % of a pound=1 oz. ? 
5. 5 score is what % of 200? 11, A 12 oz. pound is how 


6. 81% of agross = 35doz.—a#. many thousandths of an avoir- 


7, 2 of a ream of note-paper upois pound 2 


calls for what part of a thousand 12, A long T.=224°9 of a short 
envelopes. T. Express as a mixed decimal. 
197. — Sight 1. When coal is $5 a ton, what part of 
Problems. a ton is worth 40¢? $2.45? 9. At $5 a 


ton, 2400 lb. cost w 8. When a short ton is 


sold for the cost of a long ton, 24% or what per cent is gained ? 


4, 


ORAL EXERCISES 


9 is what part of 144? 


What % ? 


is x % of the number of feet in a mile. 


6, 
is notched at intervals of 5 


A centimeter = 


whole length of it ? 


7. Find the balance of this account by inspection. 


is described as Dr. and which as Cr. ? 


Vv. i BZ. Udama 


tg ; 
iin of a meter. 
centimeters. 


11% of 50 is a. 


101 
5. 1760 


A coil of wire 100 meters long 


How many notches in the 


Which party 


New York, Jan. 30, 189-. 


In acd. with B. C; D. & CO: 
Dy. Cr. 
Jan. 2| To Mdse. as by Jan. 1| By bill for services || $ 150 | 00 
bill | & 114 | 81 5| By goods returned 14 | 81 
Freight prepaid 2\13 By allowance for 
Storage of barrels 50 damages 2| 00 
16| Cash on acct. 50 | 00 


8. Multiply the sum of 11.507 and 2.09 by the difference between 
10.85 and 103. 


9, How much land can be bought for $1000.00 at 25¢ an A.? at 
$6.25? Add 183% to the total cost in each case. 


10. Of what two equal numbers less than 1 is 0.25 the product ? 
Find the square root of 0.0081. 


198. — For Read rapidly, 4, 10101 40800 935, 
Frequent «nd change to or 5, 60.004 0.048 6.6664 
Practice. the decimal 
ractice ae ss ecima 6, 39259 0.0006 833. 

, 7. 20.2020 202.020 2020.20 
eC 8, 10.12 $10.012 $112 

2. 0.311 0.0374 3.74 oe y ee ae Se 

| 63% 4% YZ th 10, sxou0 rode Teoo 


102 


DECIMALS. 


199.— Written Work; 1. 183 hundredths of $476. 


Oral Analysis. 


2. 18.75% of $476 x 100%. 


3. A city contains 40,000 persons, 26% in - 
the first ward, 32% in the second, 21% in the third, and x persons 
in the fourth. How many in each ward ? 


4) 4.5796 m. = 2.040. 
cancel. 


5. A ball containing 2.58 cu. in. weighs 4 oz. 


Give result in feet. 


Make statement and 


Give the number 


and combined weight of as many balls as contain 665.64 cu. in. 
6. 302 is the product, 174 is one factor, 100 x the other = a. 
7. 0.2 + 0.04 + 0.0001 will contain what part of 0.9 + 4? 


ty , 
8. Reduce — to thousandths. 


100 
mal ? 


one of two places. 


200.— For Dicta- 1-3. Write 
tion; Results decimally : — 
1000 1000 100 
4-6. Write as decimal frac- 
tions in lowest terms: 10% 
68% 7.3%. 
7-9. Write as decimal and as 
1i 14 22 


100 10 1000 


common fractions: 


201. — Problems 
for Statement. 


3331 


= Pp — 
1000-7 what 5-place deci 


9. 


10. Reduce 0.1875 to a complex decimal of three places. To 
To a common fraction. - 


10. Add as decimals: +5; 

11-13. A pint is what decimal 
Darby ot. ligti? lke bie 

14-15. Find the value, in lowest 
denominations, of 0.125 gal.; of 
0.375 qt. 

16-17. Give products: 11 ft. 
by 1.5; 2.162 ft. by 6. 

18-19. Give the area in sq. ft.: 
0.222 sq. yd.; 662% of 3 sq. yd. 


won 
16? 


1. A double eagle weighs 516. gr.; find the 
weight of gold that pays for 10,120 bu. of 


wheat @ 80¥. 
2. What is the rate per cwt. when the charges are $ 86.25 on 13 T., 


T., 2.0625 T., and 1.53 T. ? 


38. How many pounds.in each and all of four bins containing 
respectively 2.750 T., 23 T., 5.167 T., and 743 T. ? 


PROBLEMS. 103 


4, 240 gal. of vinegar costing 20¢ a gallon are reduced by 15% of 
water and sold at the same rate. The profit is a. 

5. 21% of a gang of 200 workmen receive $48.50. The wages of 
the rest are 20% higher. What fraction over a dollar a day does a 
workman of each class receive ? 

6. Express as a decimal part of a year: 1 mo.; 14 mo.; 1 day. 

7. What decimal part of a year has passed with Aug. 21 ? 

8, What date is 24 of a year after June 1st ? 

9, 390 books cost $54.60 to bind. What was the entire cost of 
each copy if the binding was 0.14 of it ? 

10, If 1 pound loaf of bread occupies 128 cu. in., how many equal 
1 cu. ft.? The size of 1 loaf = what part of 1 cu. ft. 


202.— Processes 1, Change ie avi ee As) of 15 =y 


with Decimals. to 128ths:— ua. esky 
21 45 39 1.54 0.15 =) of 
2. Change to 4-place deci- 7, Add decimally : — 
mals and add:— caer 4 


5 17 
ys and 34 

. Change to 100ths : — 
oi Rigen Baar ers The 3 
17) 40) 75 


4, 3,4 28, + 0.0871 


8. Reduce to lowest terms : — 
7.08 0.0708 0.70080 


Oo 


9. Write as complex deci- 


mals : — 
5. Find the difference in 4- 10.1833+ 0.87625 0.1875 
place decimals : — 
Yt? su tr) tT i18t 10, V0.6400 = a. 
203. — Problems. In solving use simple methods and few figures. 


1, The area of a floor is to the area of its 
‘supports as 10: . What area of support is given to 20 sq. ft. of 
floor ? 


104 DECIMALS. 


2. The number of feet in a mile is what per cent of the number 
of inches ? 


8. 161 ft.=1 rd. One girl lives 370 rd. from school; another 
142.35 rd. on the same road. Their houses are w ft. or y ft. apart. 


4. A boy walks 1m. and 300 steps more, each measuring 2.2 ft. 
How far in all? 


5. $81271.08 is to be divided among 7 heirs. 5 of them share 
equally ; the others receive each a double portion. What is the 
amount of a 2% tax on one of the 5 equal shares ? 


6. How much wheat at 73¢ a bushel pays for 30 sheep, in three 
lots, weighing 500, 600, and 700 Ib., if taken at the rate of $3.0174 
for each sheep on the foot ? 


7. After melting = of a sheet of metal and later =, there was 44 
of a square foot Tet How many square inches were in the et 
first melted ? 

8. If a workman saves $62.40 in a year by taking 20¢ each day 
from his wages, how long would it take 4 men at the same rate to 
save $ 124.80 ? 

9. When the cost of transporting coal is 2¢ a ton for each mile, 
and the freight on 400 tons is $200, what is the distance ? 


10. A lot of cord wood is 53 beech, 0.21875 birch, 0.1875 maple, 
= ash, 10 cd. oak, 4 poplar, iil 34% pine and fir; in all @ cd. 


204.— Mixed Ex- 1. At 1.25 cu. ft. to the bushel, compute 


amples. the value at 571” a bushel of a bin containing 
4000 eu. ft. 
9. 72.012 = 2.64 3. 70.397 -~ 0.9023 
4. Simplify wert 6. 1 sq. yd. = what part of 1 sq. rd. ? 
7 


6. 160 sq. rd/= 1A. d SQ .5r 0 1 Sq? inee=o sd. ou, 
7. From 1234,% take 347.9212. 


PROBLEMS. 105 


8, Find the profit on 9, Find the value of £23,758 
274 bbl. at $ 4.114 at $ 4.8665. 
128 “ at 3.963 10. Find the cost of express- 
if sold by the pound @ 3¢,196 ing $1,000,000 gold at ;3, of 
Ib. to the bbl. 1%. 


205.— Denominate lL. ls. =o¢. ~£1 1s. or 1 guinea’= how 
Numbers. many pence? 2 itd. or 1 farthing + 3d. 
= what part of a pound? Half a crown 

(1 crown = 5s.) 1s worth how many cents ? 

3,. Find the difference between $100 and the value of £10 10s. 
+25 M. 50 pf.+ 20 fr. 50 ct. In what (fewest) pieces of money 
could the difference be paid? 

4, $75 will buy how many pounds? The remainder will buy 
how many shillings and pence ? 

56. Add £4 6s. 9d., 21s. 43d., 5s., £4 10s. 

6. Reduce to grains: 3 sc. + 15 gr: + 4 dr. +1 oz. 

7, Find the difference between 1 lb. 5 oz. 3 dr. and 2 lb. 7 dr. 

8. At $7.00 a pound, 3 dr. of saffron cost what ? 

9, Find the value of 1000 gr. of a metal worth $20 an ounce. 

10. Three packages of opals weigh each 1 oz. 7 pwt. 5 gr. Find 
the total weight. 


206.—1. Give the latitude of the north pole; the south pole; 
the equator; a point 6913 m. south of the equator. 

2. Two persons set out from the same point. One goes 42° north, 
the other 3114 m. south. How many miles apart are they? How 
many degrees ? 

3. 1° of latitude =a m3; 1!'=y m.; 1!'=z ft. 

4, The difference in latitude of two places is 15 minutes. How 
many miles apart are they, if one is due north of the other ? 

5. A steamship laying telegraph cable finds a depth of 120 
fathoms after steaming 600 knots off shore. Give the depth in feet. 
The distance in common miles. 


106 DECIMALS. 


6. If a gold dollar weighs 25.8 gr., how many double eagles can 
be made from 3 lb. 3 oz. 372 gr. ? 

“4 gale cu, in. 1) gt. = 9 4cu.7 in. eahow snanweeaiions 
altogether can be put into two tanks holding 10 cu. ft. and 8900 
cu. In. respectively. 

8. 1 peck=acu.in. 1 qt.=ycu.in. How many 2-bushel bags 
can be emptied into a bin containing a number of cu. ft. equal to 
62 x 41 x 33°? 

9. How may 6 gal. be reduced to pints? Change 2.5 gal. to 
gills. Express 2 bu. in the largest possible units. 

10. What multipher will change tons to hundredweight ? Change 
to lower denominations: 0.875 T.; 0.625 bu.; 0.55 qt. 


207. — Interest. 1. I live in a hired house worth $ 6000. 
a General Method. For the use of the house for a year I pay 7 
(10%) of its value. The year’s rent is x dol- 
lars. This is y dollars a month, and (counting 30 days to a month) 
z dollars a day. 
2. If I had used the money which the house cost, $6000, for 
a year at the same rate, 10%, the annual interest would have been 
w dollars. For 6 mos. it would have been 4 of $2 or $y. 
3. Value of house used, $3000; rate of rent, 5%; heen rent, 
$a; 4 months’ rent, $4; 1 month’s rent, $ z. 
4. Money used, $3000; rate of interest, 5%; year’s interest, $x; 
4 months’ interest, $y; 1 month’s interest, $ z. 
5. What is the difference between rent and interest ? 
6, Interest is an allowance to the owner for the use of his money. 
The Principal is the money used. 
The Amount is the sum of interest and principal. 
The Rate of interest is the number of hundredths of the 
principal paid for a year’s use of it. 
The principal is $200. The rate is 6%. Give the interest for 
1 year; 2 yrs.; 3 yrs.; 4 yrs.; 5 yrs. The interest for1mo.? 2? 
OMe Cais eel OF Wire 8 Pins 9? eel 0 Pea tap 
7. What is a year’s interest of $300 at 2%; 8%; 4%; 5%? 


INTEREST. 107 


Find the interest What shall I pay for the use 
8. Of $300 at 4% for 2 yrs. ll. Of $1000 for 2 yrs. at 10% 
9, Of $500 at 6% for 3 yrs. 12 Of $600 for 2 yr. at 10% 
10. Of $800 at 7% for 4 yr. 138, Of $800 for $ yr. at 4% 


208.— Interest: a 1, In most business transactions 30 days 
General Method. make amonth. A month’s interest is $60; a 
Time in Months. day’s interest is $a; 10 days’ interest is $y; 

20 days’ interest is $z; ~. days’ interest is 
wren De 

2. The interest of $300 at 10% 

forl year = Bon of $300 or $a; 
for 1 month = +, of $2 or $y; 
forlday = , of $y or $z. 

8. Reckoning 30 days to an Te month gives 360 days to an 

interest year. A day’s interest is what part of a year’s interest ? 


17 days’ interest is oh of a year’s interest. 


. 4, To find the interest of $240 for 2 yr. 5 mo. at 5%. 


B. 
" f fi 29 
$240 = Principal. 29 x a a a of $249 = $ 
__.05 = Rate 
$ 12.00 = Int. for 1 yr. C. 
25, = Time in yrs. 20 
te $249 
2400 100 3 
$29.00 = Int for 2%; yrs. 12 29 
$29 


In B, what represents a year’s interest ? A month’s interest? 
2 yr. 5 mo. or 29 mo. interest? How is the process shortened? In C 
the numbers are arranged in columns for cancellation, dividends at 
the right, and divisors at the left. 


108 DECIMALS. 


5. To find the interest of $720 at 8% for 3 yr. 8 mo. 


eh 
60 a 
$129 44 of 78, of $720 = $211.20 
1 8 
1z| 44 
480 4 
3590 32 x 8% of $720 = $211.20 
§ 30 Explain each process. 


What is the interest — 
6. Of $840 for 1 yr. 9 mo. at 10% ? 
7. Of $360 for 4 yr. 10 mo. at 5% ? 
8. Of $960 for 1 yr. 8 mo. at 4% ? 
9. Of $1000 for 121 yr. at 8%? 
10. Of $400 for 2 yr. 5 mo. at 7 % ? 


209. — Interest: a 1. To find the interest of $500 for 1 mo. 
General Method. 6d. at 5%. 


Time in Days. a. 1 year’s int. = ,3, of $500, or $a. 
$ 500 b. 1 day’s int. = z1, of $2, or $y. 

10 aa mm c. 36 days’ int. = 36 x $y, or $z. 
$ 2.500 Explain the process at the left. 


2. To find the interest of $480 for 2 mo. 12 d. at 9%. 


12 
$ 430 a. 72, Of $480 = int. for . 
gre) |__ 72 c. 72X ql, of p2q of $480=int. for. 
$ 8.64 


What is the advantage of arranging dividends and divisors on 
either side of a vertical line ? 


3. The interest of $600 at 4% for 60 days. 


PROBLEMS. 109 


Find the interest of — 


4, $250 for 1 mo. 15 d. at 6% 8. $1728 for 2 mo. 17 d. at 9% 
5. $120 for 80 days at 7% 9, $800 for 9 mo. at 74% 

6. $372 for 36 days at 10% 10. $1000 for 93 d. at 44% 

7, $336 for 8 mo. 10 d. at 4% 


210. — Interest: a 1, To find the interest of $840 for 3 yr. 
General Method. Time 7 mo. 11d.at4%. (860 d. to an interest 
in Years, Months, and year.) 


Days. 2. Explain the cancellation in the process 

Process. at the left. Which numbers are then to be 

$ 840 multiphied? 38 Need they be rewritten ? 

100 A Which is used as multiplier ? 
00 08 san 4, Explain the two steps in dividing by 
104080 9000. What principle is applied ? 
9.000) $ 1092.840 5, How is the amount found ? 

$ 121.4264 


a. 3yr. 7 mo. 11 d.= 1501 d. 

b. Lyr.’s int. = 74, of $840, or $x. 
c. kday’s int. = 54, of $x or $y. 

d. 1301 day’s int. = 1801 x $y or $z. 


Compute the interest under the following conditions. Try to forecast 
the results approximately. 


Principal. Time. Rate, || Principal. Time. Rate. 

6, $270. lyr. 7mo.20d.| 6% || 14, $3863.42] lyr.5mo.12d.| 4% 
7, 500. 2yr. 8mo. 7d.| 10% |} 15, 78.30 15 mo. 15 d. 8% 
8, 810. 3yr.11mo.10d.| 4% || 16, 1566. 4 yr. 7d. 3%, 
Q, 144. Pot UM hu.) . 0%, i Le, LOU. 9 mo. 17 d. 34%, 
10, 696. 4yr. 4mo. 4d.| 3% || 18, 427.50 Pino. 12%, 
Ad © 475; lyr. 10 mo.27d.| 9% || 19, 849.78 10 mo. 29 d. 44, 
12, 84.50 |2 yr. 6mo. 15d.) 44% || 90, 648. 1 yr..1 mo. 21 d. 7%, 
18, 720. dSyr. domo. 7d.| 5% || 9], 2100. 5 yr. 5d. 5% 


110 


DEFINITIONS. 


211. DEFINITIONS. 


[FOR REFERENCE] 


Amount in computing interest. In- 
terest and principal added. 

Antecedent. The first term of a 
ratio ; the dividend. 

Bill. An itemized statement show- 
ing to whom and by whom goods have 
been sold, or services rendered, and 
giving dates, quantity, price, and 
amount. 

Common Denominator of Two 
or More Fractions. One showing 
the size of some fractional unit in 
which all may be expressed. 

Complex Fractions contain a frac- 
tion in the numerator, in the denomi- 
nator, or in both. 

Complex Decimals have a com- 
mon fraction in the numerator, as 
0.274. 

Compound Number. Two or 
more denominate numbers used to 
express one quantity ; a denominate 
number having two or more integral 
units of the same kind of measure, as 
3°°6!. 

Consequent. The second term of 
‘a ratio; the divisor. 

Couplet. The two terms of a 
ratio. 

Decimal Fractions, or Decimals. 
Any number of 10ths, 100ths, 1000ths, 
etc. ; commonly expressed at the right 
of the decimal point without written 
denominator. 

Denominate Number. One in 
which the unit is a measure, as 3 lb. 


Improper Fraction. A number 
not less than 1 expressed in the form 
of a fraction. 

Interest. An allowance to the 
owner for the use of his money. 

Invoice. A bill of goods sold. ° 

Least Common Denominator of 
Two or More Fractio1is. One show- 
ing the size of the largest fractional 
unit in which all can be expressed. 

Like Fractions have fractional 
units of the same size and kind. 

Mixed Decimal. A number con- 
sisting of an integer and a decimal 
fraction. 

Principal. A sum upon which in- 
terest may be allowed. 

Proper Fraction. A number less 
than 1; a true fraction. 

Rate of interest. Per cent of the 
principal allowed for a year’s use 
of it. 

Ratio. The relative size of two 
numbers expressed by their quotient. 

Reciprocal of a Number. 1 ~ the 
number ; the fractional unit expressed 
by that number as denominator, as 3, 4. 

Reciprocal of a Fraction. 1 - the 
fraction, or the fraction inverted. 

Simple Fraction. One _ having 
only integral terms. 

Terms of a Fraction. The numer- 
ator and denominator. 

Terms of a Ratio. The antecedent 
and consequent. 


MEASUREMENTS. 111 


Measurements. 


[Review THE TABLES ON pp. 8-9. ] 


212. — Of Lines. 1. Beginning with the shortest, name the 
five ordinary units used in measuring lines, 
or lengths, or distances. 

Compare an inch with a foot. An inch with a yard. 

Compare a foot with a yard; with a rod; with a mile. 


What part of amile is a rod? 6 What is > of a rod? 


c 


js; mM. = rods. 
$9lin.=azit.+yin. 8 3ft. x15} x 320=1m.=-2 ft. 


9. Learn the distance from home to school by measuring and 
counting your steps, or in some more exact way. 


eae I eS 


10. Estimate in rods and then in feet the dimensions of your 
schoolroom; schoolhouse; school lot; the width of the road or 
street. Test your estimates by measuring. 


213.— Length 1. 3 yd.=«@ ft.yin. 2. 621% of a mile 
Measures. = @ rods. 
At sight. 3. Ata cent a foot, 4 rods of picture cord 


will cost # cents. 
AAT 12054 yard, picture moulding for a room 265 ft. long and 
20 ft. wide will cost a. 
5. drd.=aft. 6 j-rd.=axft.oryin. 7. 100 in.=2 yd. y ft. 
gin. -8: 1000 trd.=a2m. 9. 2m. 40 rd, =-2 rd. 
10. At the rate of 3 m. an hour, how many rods can you walk 
in 15 min. ? 


214. — Length 1. An ocean steamship 660 feet long is 
Measures. what part of a mile in length? 
Written. 2. Steaming 22 miles an hour is at the rate 


of x feet every second. 3. Crossing the Atlantic, a distance of 3100 
miles, in 5 days 74 hours, the average rate is # miles per hour, 


a Bs MEASUREMENTS. 


4. Mt. Everest is said to-be 29,002 feet or a miles high. 

5. The distance from the equator to the north pole is ten million 
meters. Calling a meter 39.57 inches, what is the distance in miles ? 

6. One wheelman rides 24 miles an hour. Another rides 1 mile 
in 34 seconds. Compare the distance per minute each one rides. 

7. A race-course was 30 knots. The time of the winning* yacht 
was 3 hours 25 minutes. This was x feet per minute. 

8. A horse trotted a mile in 2.04, or at the rate of w rods and y 
feet per second. 

9. In May, 1893, the Empire State Express ran from Syracuse 
to East Buffalo, 145.6 miles, in 2 hours 21 minutes. Find the rate 
per hour. 

10. The driving-wheel of the locomotive was 64 feet in diameter. 
Calling this 5%, of the circumference, and making no allowance for 
slipping, how many revolutions would it make ? 


215. — Surface 1. What is a plane surface ? 
Measures. 2. What are the boundaries of surfaces 
[Review Tables, p.9.] Called? 38. What kind of lines bound rectilin- 
ear surfaces (rect- meaning right or straight) ? 
4. What is the shape.of most of the common units of surface 
measure? 5. Describe a square; an oblong, or rectangle. 


6. Name the five square measures of surface, beginning with the 
smallest, and giving the length of each. 7. An acre is not a square 
measure. It contains square rods. 8. 12?=a; (51)?=y; (164)’?=z. 

9. Give the length of a square yard in yards; in feet; in inches. 
10. The length of a square rod in yards; in feet; in inches. 
11. What is the length of a square mile in rods? in yards? in 
feet ? 


216.— Of Surfaces 1. Draw a diagram to show what a square 
or Areas. inchis. Isit aninchsquare? Might it con- 
tain a square inch of surface and be of some 

other shape ? 


OF SURFACES. 113 


2. Draw a diagram as an example of a square foot. If your paper 
is too small, draw it on a scale of 4 or 4 or 4; that is, make it } or 
4 or 4 of its actual length. 8. Divide your drawing to show the 
number of square inches in a square foot. How many are there ? 


4. 5. 6. 
= 3q Thi. =%. 89.10 = 84 fre Sole i 36 sq in, = 1 sq ft 
2 a 
1 1 5 
3 84 t= 47 80; in 3 84: tis ence Lh 60 sq. in. or sq. ft. 
; Beit. ==4y BQ4 10, Bde 2.50. tt | LOS Sq I =3 sq. ft. 


7. 662% of a square foot =x sq.in. 8. 14 sq. ft. = sq. in. 
9. Represent a square yard in outline; scale, 4. Separate it into 
square feet. 10. How many square inches in it? 


217. — Of Land LL roa yeaa ott. 2 i. 
Areas. 2. Draw a diagram on a scale of =; (44 of z 
Written. inches long) to represent a square rod. From 


one corner mark the yards along two sides. 
3. Separate it into square yards. You find that you have 25 squares, 
a half-squares, and y quarter-squares. 

4. If convenient, outline a square rod on the schoolroom floor; 
imagine one on the ceiling, and show how far it would extend; or 
have one marked off in the school-yard. 

5. How many square yards in an acre of land? 6. Draw a fig- 
ure to represent an acre, 10 rods wide and 16 rods long; scale, + in. 
to arod. Divide your drawing to represent square rods. 

7. What part of an acre does the school lot equal? 8 7A.=2 
sq. rd. 

9. How many acres in a square mile? In 10% of it? In } 
of it ? 

10. A western township is 6 miles square. It contains w square 
miles or sections, and the distance around it is y miles. 


11. Find some piece of ground which you can show to contain 
about 1 A. 


114 MEASUREMENTS. 


218. — Surface 1. Change 20,000 sq. in. to square feet. 
Measures. 2. Change 12,371 sq. ft. to square rods. 
Written. 3. Change 287 sq. rd. to square feet. 


4. 1350 sq.m.=a A. 65. Change an acre to square feet. 

6. Bought 4 A. for $400, and sold it at a dime per square foot. 
How much did I gain or lose ? 

7. A farmer owns five fields or lots measuring as follows: 80 A., 
200 sq. rd.; 2 sq. m.; 874% of an acre; and 435,600 sq. ft. What 
is the acreage of this farm ? 

8. In 20,000 sq. ft. how many square rods ? 

Ode: 10. What will be the cost of a 

17 sq. ft. 19 sq.in. school lot containing 32,670 sq. ft. 
19 sq.ft. 75 sq: in. at $5000 per acre and $12.50 per 
42 sq. ft. 108 sq. in. — square rod for filling and levelling ? 
96 sq. ft. 121 sq. in. 


219. — Lines. 1. Draw horizontal, vertical, and inclined 

or oblique lines. 2. Name the three kinds 

that you have drawn and describe them according to their direction. 

Try to make your description exact and brief before consulting 
page 142. 

3. With reference to each other, two lines may be parallel or per- 
pendicular. How many pairs of parallel lines on pages 116-117 ? 
4. What are parallel lines ? 

5. Describe straight lines and curved lines. 6. What is a line? 


220. — Angles. 1. Draw an angle and show its sides and 
vertex, or their point of meeting. 

2. Two lines having different directions and 
meeting at a point make an angle. To measure 
an angle is to measure this difference in the 
direction of the lines. Repeat the table for 
circular measure (p. 9). 


3. If you prolong the sides of an angle, do 
you increase its size ? 


OF ARCS AND ANGLES. 115 


4. Draw two intersecting lines so as to make four equal angles. 
5. The lines thus drawn are perpendicular to each other and the 
angles are right angles. Define perpendicular lines. 6. What is a 
right angle ? 

7. Draw two inclined lines perpendicular to each other. 


8. Compared with right angles what are acute angles? Obtuse 
angles? 9. Show how many of each of the three kinds are on 
pages 116 and 117. 


10. Adjacent angles have one side in common. With two strokes 
of a pencil draw four angles; then draw four figures showing the 
four pairs of adjacent angles that you have made. 


11. Unequal adjacent angles are oblique angles. “Oblique” 
means ~ ? 


221.— Divisions of 1. What is a circle? 2. What name is 
Circle; Degrees of given to the curve that bounds it? 3. A 
Arcs and Angles, diameter of a circle bisects it. How is the 

Oral. diameter indicated in the figure? 4. How 
many radii are drawn ? 

5. The surface enclosed by bca is a quad- 
rant; by bcg, a sextant. These same names 
are apphed to the arcs ba and bg, as the cir- 
cumference is often called a circle. What 
part of a circle or circumference is bea or ba? 
beg or bg? 


Oo 
ts) 


g f 6. For convenience in measuring arcs and 

angles, every circumference, whether large or 

small, is divided into 360 equal parts or degrees (360°). How many 

degrees is a semicircle? a quadrant? a sextant? a sign or 12th of 
a circle ? 


7. Each degree is divided into 60 minutes (60'), and each minute 
into 60 seconds (60"). 
1S = wy! . 300! = w® An are of 30° contains y/ 
Baek Or == oi 600" = a! 8 of a circle contains 2° 


116 MEASUREMENTS. 


9. Over how many degrees does the long hand of a watch move 
inan hour? in 80 min.? nth.? 10. In 20 min.? in 25 min.? 
in 85 min. ? in 4 day? 

ll. Of the six angles in the figure which is the right angle ? How 
many degrees in the arc between its sides? What arc measures a 
right angle ? 

12. Which three angles in the figure are equal? What is the 
size of the arc that measures each? Each is an angle of a. 
13. How many degrees in half a right angle? 

14. Draw angles of 90°; 45°; 60°; 105°; 120°. Which are obtuse ? 
Which acute ? 


222.— The Six 1. Notice how many sides these figures 
Quadrilaterals. have, and define a quadrilateral. 
Oral. 2. Which have their opposite sides paral- 


lel? What is a parallelogram. 
8. Which of the quadrilaterals has only 


l two parallel sides? What is a trapezoid? 
4. Which is a quadrilateral without paral- 
an : lel sides? What is a trapezium ? 
5. Which two parallelograms are equilat- 
™ 0 eral? LEquiangular? Rectangular ? 
l 


6. What name is given to a quadrilateral 
with four right angles? 7. To an equilateral 
rectangle ? 

8. Which of the parallelograms have only 
oblique angles ? What is a rhomboid ? 

l 9. Which rhomboid is equilateral? What 
is a rhombus ? 


3 
ic) 


10. Show the propriety of each of the 
following terms as applied to A: quadrilat- 
eral, parallelogram, rectangle, square, equi- 
angular, equilateral. 


3 
FS 
9 


QUADRILATERALS. 117 


11..Of each parallelogram which line is 


7 
the base 2? 12. Which shows the altitude or 
height of the parallelogram ? 


m7 re 18. The altitude and base must always be 
perpendicular to each other. Try to tell why. 


l 
14. A straight line, like Jo, that joins the 
vertices of opposite angles is a diagonal. 
Prove by cutting — 
‘ (1) that a diagonal bisects a parallelogram. 
(2) that the opposite angles of a rhomboid 
are equal; and by measuring prove, 
(3) that the sum of the angles of a quad- 
rilateral equals four right angles. 
0 


223.— Of Rectangles. 1. An inch-wide rectangle 12 in. long con- 
tains wsq.in.; a 12-inch-wide rectangle of the 
same length contains 12 x @ sq. in., or y sq. 

in. (See the figure.) 


2. What is the area of a rectangle 15 in. 
square? 8. Of a rectangle 1 ft. 6 in. square? 


4. A foot-wide rectangle 24 ft. long contains 
# sq. ft.; a 12-ft.-wide rectangle of the same 
length contains 12 x x sq. ft. or 
y sq. ft. 

5. Find the area of rectangles 
15 ft. long, ‘7 ft. wide; 13 it. x 
20 ft.; 164 ft. long, 164 ft. wide; 
5i yd. x 54 yd. 


@ = : 
ST Ss«*6. A piece of land measures 
yg 25 rods one way and 20 rods 


the other way. Find its area in square rods; in acres. Which of 


118 MEASUREMENTS. 


the following statements or equations is right for the second 
answer ? 
25 x 20 sq. rd. cL 25 x 20 sq. rd. 
160 160isq;.50 ee 
7. A kindergarten table 4 ft. 3 in. long and 20 in. wide is marked 
off into inch squares; how many are there ? 
8. How many square yards in a web of cotton 404 yd. long and 
3 ft. wide ? 
9. In a flag 103 ft. long and 2 as wide, how many square yards 
of bunting, not allowing for seams ? 
10. A patchwork quilt measuring 3 yd. by 3 yd. is made of 4-inch 
silk squares; how many are there ? 


224. — Superficial 1. What shall I pay Mr. Bates for concret- 


Contents of ing a walk 60 ft. long, 4 ft. wide for half its 

Rectangles. length, and 3 ft. wide the rest of the way ? 
Written. His price is 75 cents per square yard. 

Norr. — Draw diagrams to 2. Mr. Cross fenced his strawberry patch, 


illustrate ; make statements, 
and cancel when possible. 


which was 4 rods wide and 100 feet long, 
with three lines of barbed wire at 11% per 
foot. The posts cost $7. How many quarts of berries at 5f a 
quart must he sell to pay for the fence ? 

3. When an acre of land is 40 rods long, what is its width ? 

4. Mrs. Fiske bought the equivalent of a square yard of 4-inch 
ribbon for $4.50; what was the price per running yard ? 

5. How many square tiles 9 inches long will lay a floor 12 ft. 
wide and 27 ft. long ? 

6. A roll of oilcloth 72 in. wide is 30 ft. long ; what is it worth 
at 621¢ per square yard? 

7. How many square feet of glass in your schoolroom windows ? 
8. Of blackboard surface in the room ? 

9. Drawing paper measuring 24 x 36 is cut into 9 x 12 pieces. 
How many pieces will a ream furnish ? 


OF FLOORS. 119 


225.— Of Carpet- 1. Ingrain carpets are generally woven in 

ing, Tiling, etc. strips a yard wide; other carpets, three- 
quarters of a yard. What two advantages 
come from running the strips lengthwise of 
the floor rather than across it ? 


Oral and written. 


2. On floors of the following widths which width of carpet could 
be used without either cutting or turning under any strip ? 


12ft. 15ft. 22ift 227i. 13ift. 20ft. 18 ft. 


3. How many strips of ingrain carpet will be needed for a room 
18 ft. square? How many running yards? How many yards must 
be bought if a quarter-yard is wasted in matching each two strips ? 


4. How many strips of brussels or tapestry carpet will be needed 
for a room -15 ft. wide and 21 ft. long? How many yards, if it 
matches without waste? Find the cost at $1.25 a yard. 

5. Find the cost of covering a floor 14 ft. by 20 ft. with yard- 
wide carpet at 75%, no strips to be cut, nor allowance made for 
waste. 6. What will it cost using three-quarter carpet at $1.50, on 
the same conditions? 7. Using 4-ft. oilcloth at $1? 


Find the cost of carpeting floors under the following conditions (strips 
that are cut cost as if whole) :— 


Length of Width of Width of Allowance for Cost 
room. room. carpet. matching. per yd. 
els. ff: 14 ft. 1 yard 13 yds. — $0.90 
9. 22 ft. 18 ft. 3 yard 21 yds. 4.25 
Mose rior tip! 182 ft. 2 dyad t yd. 0.874 
Tree th 20 e 8 yard 23 yds. 1.374 


12. How many 8-in. marble tiles are required to cover a hearth 
2 ft. by 4 ft. 8in.? 18. To cover a floor 20 ft. by 464 ft. ? 


14. The areas to be tiled about a fireplace are: one of 5 ft. 3 in. 
by L.ft. 9 in.; two of 2 ft. 104 in. by 1 ft. 9 in.; one of 1 ft. 9 in. 
by 6 in.; one of 5 ft. 3 in. by 2 ft. 44 in.. Find the total area. 


16. How many tiles 1} in. by 3 in. are required ? 


120 MEASUREMENTS. 


16. Find the area of the surfaces shown 
at. the left. 17. How many 2-in. tiles are 
required to cover them? [°, ', p. 148.] 


18. A room is 134 by 18, and 8 ft. high, 
How many rolls of wall paper are required, 
each being 8 yd. by 18 in., no allowance 
made for doors, windows, or baseboards ? 


226.— Of Roofs, 1. 93 squares of slating are required to 

Pavements, etc. cover a certain roof. This is equal to how 

many square yards? If the slates are 8 x 16, 

and each course overlaps 10 inches of the one below it, find the 
number of slates used. 


2. How many blocks 6 x 4 inches will be used in paving a four- 
rod square ? 

38. How many tin plates 13 x 19 must be used for 1 square of 
roofing, if they are lapped or folded 4 in. 
on each side ? 

4. Three piazza roofs about a house 
measure in feet 308, 24x7, 74123. 
How much less than 5 squares do they 
contain ? 


5. A house lot contains + A. How 
many sq. ft.? The house is 274 x 40. 
What would it cost to sod the remainder 
at $1.50 a square ? 


6. Let this figure represent the out- 
line of a cellar. Copy, and divide it 
into 5 rectangles. From the given 
dimensions find those of each rectangle. 
7. How many square yards of cement 
would be required to cover the bottom of the cellar ? | 


OF RECTANGLES. 121 


227.— Areas and 1. A chess-board contains 64 squares 14 in. 

Perimeters of long. What is its perimeter? If it has an 
Rectangles. inch-wide border, what ? 

For oral analysis. 2. In a 2-inch square how many 14-inch 


squares ? How many +-inch ? 


3. Compare the perimeter of a 4-foot square and an equal surface 
8 feet long. 


4. My sidewalk is 10 ft. wide besides the curb, and 100 ft. long. 
How many 4 x 8 bricks in it? 


5. Compare a 4inch square and a 12-inch square as to length 
and area. 


6. A marble slab 4 ft. by 21 ft. was sold for $4.50; price per 
square foot ? 


7. What is the area of a square that can be set off with 200 feet 
of rope ? 


8. How many boards 9 inches wide make a close fence 8 feet high 
around three sides of a square lot 180 feet long ? 


9. A hall measures 12 feet by 36 feet. How many breadths of 
yard-wide carpet would be needed? How many yards, allowing 
3 yards wasted in matching the pattern ? 


10. A room 14 ft. by 18 ft. is to be covered with yard-wide carpet 
at $1. Which is the cheaper way to run the strips? Why? 


228.— Of CityLand. 1. Mr. Sharp bought land bordering on 

Written. Spring Street between Poplar and Maple at 

[See next page.} 3 cents per square foot, which he cut up into 

building lots. He first laid out a 40-foot street in the rear, which 
he called Leland Street. What did he pay for the land? 


2. He sold lot C to a civil engineer for his services in surveying 
and making plans, plus an additional 2 cents per square foot. What 
did the survey, ete., cost ? 


122 MEASUREMENTS. 


After reserving lot A for his own dwelling-house, he sold the 
remaining 10 lots by auction at the following prices. 


Be ee ee ee ee 


Spring 


Poplar 
Maple 


2 Leland 


as TS Se ay ar 


Find the proceeds of the sale of each lot. 
8. Lot B for12i¢ 6. Lot F for183¢ 10. Lot J for 193¢ 
4. Lot D for 15¢ 7. Lot G for 21¢ 11. Lot K for 173¢ 
5. Lot E for 223¢ 8. Lot H for 201 12. Lot L for 251? 
9, Lot I for 173¢ 
13. Before the sale, he opened and laid out a 16-ft. alley from 
Maple Street to Leland. What did the alley cost him, $85 being 
paid for labor ? 


OF RHOMBOIDS. 123 


14. The grading of Leland Street cost him $3.75 per square rod. 
What did the street cost, including land and labor ? 


15. Mr. Sharp laid a sidewalk 8 ft. 8 in. wide on two sides of his 
own lot A. The 8-inch edge-stones cost him 80 cents per running 
foot. The brick cost $12 per thousand, and the labor $58.25. The 
bricks were 8 x 4 x 2 and laid flat. What did his walk cost? (Make 
statement. ) 

16. The owner of lot I paid an average of $3.50 per rod for fenc- 
ing. It cost him $2 if he paid for only half of the division fence. 

17. The abuttors combined, and concreted the alley at 561 cents 
per square yard. What was the total cost? 18. What part of the 
whole cost should the owner of J pay? 19. What is the assessment 
of the owner of I? 

20. Leland Street is accepted by the city and paved at a cost of 
$3 per square yard, the abuttors agreeing to pay 25% of the cost of 
the part adjoining their property. What is the assessment on lot L? 


229. — Of Rhomboids. 1. Cut a rhomboid from paper. 

2. Divide it along any altitude line. 

3. Adjust its parts so as to form a 
rectangle. 

4. Compare the base and altitude of 
the rectangle with the base and altitude 
of the rhomboid. 

5. How is the area of the rectangle 
found ? 

6. How, then, may the area of the 
equivalent rhomboid be found? Area = 
base x altitude. 

Find areas of rhomboids with — 

7. Base 122 ft., altitude 74 ft. 

8. Base 16 rd., altitude 40 ft. 

Ae As Faas Ce ai Re let ve 

10. A = 83 ft., B 54 in. 


124 MEASUREMENTS. 


230. — Of Trapezoids. 


Oral. 


1. Cut out a trapezoid 

having two right angles. 

2. Divide it along a middle line parallel to its parallel sides. 

3. Adjust the parts so as to make a rectangle. Notice where the 
parallel sides of the trapezoid are to be found 
in the rectangle. 

4. Compare the area of the rectangle with 
the area of the trapezoid. 

5. Show that one dimension of the rectangle 
equals the sum of the parallel sides of the 
trapezoid. 

6. Show that the other dimension of the 
rectangle equals one-half the altitude (length 
or width) of the trapezoid. 

7. How is the area of the rectangle found ? 

8. The base and altitude of the rectangle 

correspond to what lines in the trapezoid ? 

9. How, then, may the area of the trapezoid be found ? 


231.— Of Trapezoids. At Sight. —1. In connection with this 
trapezoid explain this statement: 
(12 + 16) sq. in. x $=112 sq. in. 

2. Show that the average or mean 
or middle length of the trapezoid is 
(12 + 16) + 2. 

8. Is there any difference in value 
eres x8, 02416) x5 


oF a2) x8» 


between 


OF TRAPEZOIDS. 125 


4. In what three ways may you state the process of finding the 
area of a trapezoid ? 
5. What is the altitude of trapezoid B? ithe sum 
of its parallel sides, or mean width? Explain - 
8 + 16 
2 
(mean length x width, or) _ 
(mean width x length ; 


x 56 =o. 


6. In trapezoids 


Written. — Find the area of trapezoids measuring — 
7. Parallel sides 16 ft. and 24 ft.; altitude 13 ft. 
8. Parallel sides 25 in. and 24 in.; altitude 44 in. 
9. Parallel ends 13 in. and 16 in.; length 14 ft. 


10. A trapezoidal board is 74 in. wide in the middle and 16} ft. 
long. 


232. — Of Oblique- 1. Cut an oblique-angled trapezoid along 
Angled Trapezoids. its middle line and place 

its parts end to end to 

form a rhomboid. 


2. What lines of the trapezoid form base and 
altitude of the rhomboid ? 
3. How may the area of the rhomboid be 
found ? 
4. Show how an oblique-angled trapezoid may 
be changed to an equivalent rectangle. 
5. Find the area of a trapezoid measuring 223 
ft. in altitude, 723 ft. and 853 ft. along its paral- 
lel sides. 
6. A 10-ft. wall is 128 ft. on the ground and 122 ft. along the top. 
What will it cost to paint both sides at 9% a square yard ? 
7. Draw a rectangle to represent the area painted. (Scale, ;,/55) 


126 


233. — Of Triangles. 
Written 


re 


MEASUREMENTS. 


1. Show by measuring with a protractor, or 
by cutting and laying the angles together, that 
the sum of the angles of a triangle equals two 
right angles (180°). 

2, How many right angles may a triangle 
contain? How many obtuse? How many of 
60° ? 

3. Find the size of the third angle when 
two angles of a triangle measure 90°, 30°; 
120°, 40°; 65°, 35°; 624°, 874°. 

4, Triangles are named from their largest 
angles, — right, obtuse, and acute. How many 
of each kind are represented here ? 


5. Named from their sides, triangles are 


equilateral (three sides equal), isosceles (two 


sides equal), or scalene (three sides unequal). 
Give examples of each from the drawings. 


6. Cut from paper any one of the four par- 
allelograms (pp. 116, 117) and find its area. 

7. Bisect it along one of its diagonals. 
What is the area of the resulting triangle ? 

8. Compare the base and altitude of the 
parallelogram with those of the triangle. 

9, How is the area of the parallelogram 
found? The area of the triangle ? 

Find areas of triangles of the following 
dimensions : — 

10. Base = 40, alt. = 18. 


1]. Base = 60, alt. = 25. 


VIB Sit Balt. =.9 i, 137 3B 4 rh ali it. 


14, 191 ft., 21 yd. 


16. Show that B x } 


15. 38 rd., 224 ft. 
alt., Bx alt., or 4(B x alt.) = 


OF CIRCLES. 12a 


234.— Of Trapeziums, 1. Draw or cut out a trapezium. 
Written. 2. Separate it into two triangles along one 
of its diagonals, as lo. 
3. Find the dimensions of each tri- 
angle and its area. 
4, What will the area of the trape- 
zium be ? 
5. The diagonal of a trapezium is 
24 inches; the altitudes perpendicular 
2 toit are 18 inches and 15 inches, re- 
spectively. What is its area ? 
6. The diagonals of a trapezium cross at right angles. The point 
of intersection is 50 feet from the upper end of each diagonal. One 
diagonal is 100 feet long, and the other 150 feet. Find the area. 


235.— Of Circles: 1. Bring to school the results of very care- 
Diameters and fully measuring the diameter and circumfer- 
Circumferences. ence of several circular objects: plates, rings, 
Oral and written. covers, wheels, or coins. 


2. In each case divide the circumference by 
the diameter, carrying the division to several 
decimal places, and compare the quotients. 


3. If you have measured and divided accu- 
rately, the quotient in each case will be 3.1416—-. 
What does this show ? 
4, In like manner divide your diameters by 
the corresponding circumferences. Your quotient 
should always be 0.31831. What does this show ? 
5. How many diameters make a circumference ? 
6. What part of a circumference equals a diameter ? 


7. The diameter of a circle is 10 ft.; the circumference = 3.1416 
x 10 ft., or x ft. 


128 MEASUREMENTS. 


8. The circumference of a circle is 10 ft.; the diameter = 0.31831 
GLO ity or eT 
9. Compare 10 + 5.1416 and 0.31831 x 10. Which is easier, to 
divide by 3.1416 or to multiply by 0.31831 ? 
10. 3.1416 = the ratio of the circumference to the diameter. It is 
represented by the Greek letter 7 (English p). D= diameter; 
C= circumference; A= radius. Interpret the following: — 


C=) xr DO; T _ 0.31831; D= Gxe 
Tv Tv 
Find the diameter or circumference or radius. Forecast the result. 
1 Wp #7 1b reed CS 146 C= F402 San. ee 
TAP (Ch 20D ith ae) ea 16.2 Da 167 an Cae 
ISR == 90 Tee Ce. 167 C= iy dy ets hee 
236.— Circles With Objects. —1. Cut a circle from paper. 
changed to Hqui- Bisect it and eut each half into fourths. 
valent Rectangles. 2. Arrange the eight sectors as in B, forming 


a figure somewhat lke a rhomboid. 


3. Cut another equal circle, divide 
it into sixteen equal sectors, and ad- — 
just them to form a figure still more 

eae like a rhomboid. 
os 4. Of the two rhomboids, which is 
What 


more nearly a rectangle ? 
part of the circle is its base? Its 
altitude ? 


5. Imagine a circle cut into a 

thousand equal sectors, arranged as 

B before. Would the figure formed be 
a rhomboid or a rectangle? 6. Com- 

pare its base with a straight line. 

What can be done to make its base 

more nearly straight? 7. The circle 

C would then be changed to a rectangle 


OF CIRCLES. 129 


having a base equal to . and 
an altitude equal to ... How 
would the area of this rec- 
tangle be found ? 


8. Explain the figure D. 


9. Interpret the equation: 


D Area of circle = R x 5 


ae, es es C= ahout2o «Ai. 

Find the area of circles : — 

ll. Radius, 6 ft.; Cir. x 13. Cir. 100 ft.; Diameter, x 
12. Diameter, 10 ft.; Cir. 14. Cir. 50 ft.; Radius, x 


237.— Diameter of Oral. —1. efgh is the square of the di- 
Circle Given, to find ameter of the circle. A circle is what part 
the Area. of the square of its diameter ? 


2. Explain the following (as shown 
in §§ 255-6) : — 
(a) A=Dx &; but C= D x 3.1416; 
hence, 


(b) va gaa Bp 


2X a22* ; cancelling, 
we have 

(oye A176) 50.7804; bute Dx 
D=D*; hence 

(dq) A= D’ x 0.7854; that is, a circle 
is 0.7854 of the square of its diameter. 

3. Inthe figures at the left the shaded 
portion is the area of the circle. What 
decimal part of the square is it ? 

What decimal part of the square is 
not shaded ? 


130 MEASUREMENTS. 


4. A circle is what part of its circumscribed square ? 
5. How is the area of a circle found from its diameter ? 
Written. —6. What is the area of a 5-foot circle ? 
7. What part of 3.1416 is 0.7854 ? 
8. Find the area of a 12-inch circle. 
9. What is the cost of a circular piece of aluminum at 30 cents 
per square inch, the radius being 3 inches ? 
10. What is the area of a circular pond 200 feet in diameter ? 


238.— The Area of 1. In the following figure we have three 
Circles. rectangles each equivalent to the circle. 


Which one is twice as long and half as wide as H? Which is half 
as long and twice as wide as H? 

2. The length of each is what part of the circumference ? 

3. The width of each is what part of the diameter ? 

4. The diameter of a circle is 10 feet. Whatis its area? Explain 
each of these three methods of finding it. 


0.7854 
[ Rect. H.|] — a ae # poe) = 78.54. Formula: R x c= A. 


1 0.7854 R 
[Rect. 1] x (10 x 3.1418) = 78.54. Formula: > x C= A. 


0.7854 


yl 
[Rect. J.] 10 x (Sree) = 78.54.. Formula: D x — A, 


PROBLEMS. 131 


5. The circumference of a circle is 12 feet. Explain this method 
of finding its area: — 
C = 12 feet. D = 0.381831 of 12 feet. 
3 
(0.31831 x 12) x i= 11.45916. 


Written. — Find the area of circles when — 
6. Diameter = 40 8. Radius = 24 10. C = 200 
7. Circumference = 80 9. Diameter = 36 We Des bo 


239.— Oral Review. 1. What objects before you are nearest in 
length to a yard? to a foot? to a rod? 
2. How many degrees measure aL? Can you find as you look 
about any except right angles ? 
3. After going 1 round the world, «° complete the circuit. 
4. If the angles of aA=2L/’s, each angle in an equilateral A 
measures @°. 
5. How may the area of a triangular park be found ? 
6. How much of an 8-in. square is not covered by a 7-in. square ? 
by a 6-in. one ? 
7. A rhombus containing 3 sq. yd. contains how many sq. ft. ? 
8. Which measures more, a rhombus or a square with the same 
perimeter ? 
9. The top of a round table has what RE of the area of that of 
a square table of the same diameter ? 
10. A triangle = 4 the area of _. 


240.—Drawingand 1. Draw a 2-in. circle; then the largest pos- 
Measuring of Figures. sible square inside, and one of its diagonals. 
2. Calling the diagonal the base of the tri- 
angles, what is the area of each? 
3. The area of the largest square drawn in a 3-in. circle = a. 
4. Draw a 1-in. square and a 1-in. rhombus. Are their bases the 
same? ‘Their altitudes? ‘Their areas ? 


abs y: MEASUREMENTS. 


ol 


5. Using 1-in. lines, make a rhombus with an altitude of + in. 


Its area will be —.. 


tole 


6. Using 1-in. lines, make a figure of as small area as you can. 
What is its shape ? 

7. As a ring is flattened does its capacity change? Does the 
length of its perimeter ? | 

8. Draw a trapezoid; the horizontal and vertical lines may be 
1, 14, and 2 inches. Divide it into a rectangle and a triangle. Find 
the areas of each, and add; then find the area of the trapezoid in 
the usual way. 


9. Draw a rectangle and a second figure with the same length of 
line, but noL’s. What is its shape? How do the two differ in area ? 


10. Explain what dimensions you need to know in order to find 
the area of a trapezium that you have drawn. 


241.— Problems in 1. A button is 4.7124 in. round. How long 
Measuring Circles. a button-hole is required ? 
Written. 2. Find the circumference of the base of a 
lamp chimney that is 23 in. across. 
3. A circus ring is 414,45 ft. round. Find the distance to the 
centre in rods. 
4. A hogshead is a little over 121 ft. round the middle. Will 
it go through a doorway that is 3 ft. 10 in. ? 
5. If a mountain is 10 m. round, what distance might be saved 
by tunnelling ? 
6. A pie is cut accurately into 6 equal pieces. Which is longer, 
the curved edge or the straight one ? 
7. The hubs of two wheels are alike, but the spokes of one are 
3 in. longer. How much greater is its circumference ? 
8. If a barrel is 18 in. over the chine, how much strap iron will 
be required to make 100 end-hoops with 3-in. laps? Make a state- 
ment. 


OF SOLIDS. 133 


9. In a lawn 100 ft. square the circular basin of a fountain is 
40 ft. from each side. Draw a figure, and find the area of the 
greensward. 

10. When you know the area of a circle, how can you find the 
radius ? 


242. — Of Solids. 1. Lines have one dimension; viz. W. 
y] 
[See p. 9.] 2. Surfaces have two dimensions; viz. 
OT wets 


3. Solids occupy space and have three dimensions, viz. —, ~, 
BuO 2 

4. Mention the three common solid measures in the order of their 
size. 6. Compare each one with the one next larger or smaller. 

6.-10 cu. ft. = 2 cu. in. 8. 10 cords contain x cu. ft. 

feat curve 7 CU. Lt 925°720 cunits=—& cu. yd. 

10. State the method of finding the number of cubic feet in 20,000 

cubic inches. 


243. — Of Cubes. Oral. —1. What is arectangle? 2. A solid 
bounded by six rectangles is a rectangular solid. 


3. A cube is a solid with six square faces. How many corners, 
edges, angles, has a cube ? 
4. Is acommon brick a cube? Is it a rectangular solid? What 
is an equilateral rectangular solid ? 
5. Describe an inch-cube, or a cubic inch ; 
a cubic foot; a cubic yard; a 2-foot cube. 


6. What is a 9-inch cube? How many 
cubic inches in a 9-inch cube ? 

[See the figure.] Along one edge of a 
cube there is a row of x cu. in.; 9 such 
rows make a tier of 9X cu. in. or y 
cu. in.; 9 such tiers contain 9 x y cu. in. 
or 729 cu. in. 

DiMemetiaae x9 % 9 CU. in, = 2 Cu, 10, 


134 | MEASUREMENTS. 


Written. — In a similar or better way show the contents of — 

7. A 6-in. cube 9. A 5-foot cube ll. A 10-yard cube 
8. An 8-in. cube 10. A 12-foot cube 12. A 20-in. cube 

TS MD tied , pO Am Cle O° et Usa oe ames, 

14. V/64? 1/216? 1728? W512? ¥/1331? 729? ~/27000? 


244. — Rectangular 1. Prisms are named from their bases, — 

Prisms. square, rectangular, triangular, hexagonal, ete. 

_ Name some familiar objects that are rectangu- 

lar prisms; that are square prisms. Which kind includes the other ? 

2. Find the cubical contents of a rectangular prism 
whose dimensions are 9 in., 4 in., and 5 in. 


Notice in the figure the number of cubic inches in 
a row, the number of rows in a tier or layer, and 
the number of tiers‘in the prism; and then explain 
the statement: 9 x 4 x 5 cu. in=~# cu. in. 


3. How many inch cubes may be put into a box 
10 in. long, 8 in. wide, and 5 in. deep ? 

4. A trunk measures 3 ft., 20 in., and 18 in. Find 
its cubical contents. Why not multiply 3 by 20 by 18 instead of 
36 by 20 by 18 ? 


Find the contents of rectangular prisms of these dimensions : — 


Length. Width. Height. Length. Breadth. Depth. © 
Deel Ont. cel 4G eo ts 8. 421 ft. 20 ft. 13% ft. 
TEs Ras Ra ade 8 ES Pinal 9. 12iyd. 10ft. 16in. 
(Fare harem sega 2 cok 10. 204 ft. 172 ft.) Orin 


245.— Of Cord- 1. Wood for fuel, sold by the cord, is usually 
Wood. in sticks of what length? 2. In what form 
Oral and written. are they piled to make a cord? 8. Give the 
dimensions of a cord; a half-cord; 3 of a cord 

or a cord foot. 


OF WOOD. 135 

4. What kind of solid does a half-cord resemble? 6. Explain: 
8x4x4cu. ft.=2 cu. ft. as applied to cord-wood. 

6. A pile of 4-foot wood, 4 ft. high and 8 ft. long, contains a cord. 
Tf16 ft. long? 24ft.? 32 ft.? 96 ft.? 

7. A pile of 4-ft. wood of the usual height must be how long to 


~ 


contain 10 cords? 12 cords? 25 cords? 


8. How many cords in a pile of 4-ft. wood, 4 ft. high and 18 ft. 
long? Explain the following statement, and show what may be 


cancelled : — 
4x4x 18 cu. ft. 
128 cu. ft. 
9. Bought a pile of 4-foot wood 30 ft. long and 8 ft. high at $6 
per cord. 
4x8 x 30 
128 
In the statement what represents the number of cubic feet? The 
number of cords? The cost of all ? 


x $6 =2. 


Find the value of piles of wood, as follows : — 
Length. Width. Height. Price. Length. Width. Height. Price. 
10. 24 ft. 4f. 6ft $4. 13. 24 ft. 4ft. Tift. $3.50 
Pisa te 5G: TLS Te. oD. 14. 20 ft. 3 ft. 12 ft. 5.00 
Bee RO Ste 10 fer TE: 6. 16. 163 ft. 44in. 22ft. 4.25 


246. — Of Lumber. 1. Timber sawed for building purposes is 
Written. lumber. What forms can you mention besides 
boards, planks, joists, and beams ? 

2. In measuring lumber no attention is paid to the thickness 
unless it exceeds an inch. A board 12 ft. long and 12 in. wide 
and 1 inch or less in thickness contains 12 sq. ft.; if 10 in. wide, 
it contains 19 or 2 of 12 sq. ft. or = a sq. ft. 

3. A board 15 ft. long, 8 in. wide, and # in. thick contains how 
many square feet ? 


Nore. —If 15 ft. long and ove foot wide, it would contain @ sq. ft.; being % of a foot wide, it will 
contain, etc. 


136 MEASUREMENTS. 


4. 10 16-ft. boards averaging 9 in. in width contain a square feet. 
Explain: 10 x 2 of 16 sq. ft. 


5. A board 1 inch thick and a foot square is 
a board foot. « of them piled together would 
make a cubic foot. 
6. A board 10 ft. long, 12 in. wide, and 1 
inch thick contains 10 bd. ft. If 2 in. thick, it 
would contain 2 x as many bd. ft. ora. If 14 
in. thick? If 14 in. thick? If 24 in. thick? 
If a ft. thick ? 
7. Find the contents of a 3-in. plank 15 ft. long and 10 in. wide. 
Explain: 3 x 15 x 2=~@. 
8. 12 joists, 16 ft. long and 4 in. square, contain & board feet. 


Find the contents in board feet of lumber measuring as follows : — 
9. 6 boards, 16 ft. long, 14 thick; width in inches: 8, 10, 12, 13, 
14, 9. 
10. Fifteen 3 x 5 joists, 18 ft. long. 
11. A stick of timber 18 ft. long and 12 in. square. 


247.—To find the 1. How many faces has a rectangular 
Surface of a Cube. prism? 2. What name is given to a rectan- 

gular prism when all its faces are equal ? 
Megat 3. Find the entire surface of a 5-inch cube. 
Explain the statement: 5? x 6 = 2. 


4. The entire surface of a cube is 150 sq. in. How long is the 


cube? Explain the statement: (aes a (eh 


Find the entire surface of — How long a cube has— 
5. A 9-in. cube 8. An entire surface of 384 sq. in. ? 
6. A cube 10 in. long 9. An entire surface of 600 sq. ft. ? 


7. A 16-in. cube 10. An entire surface of 294 sq. in. ? 


OF PRISMS. 137 


\ 


248.— Of Rec- 1. Compare with each other the ends of 
tangular Prisms. a square prism. 2. Compare its four sides. 
Written. 3. Find the entire surface of a square 
prism 8 in. fong and 3 in. wide. Ex- 
plain the equation : — 
2x(8x3)+4x(8 x 3)=@. 

4. Compare the opposite faces of 
any rectangular prism. 9. Find the 
entire surface of a prism measuring 
Gby 4 by 2. Explain the statement : — 


z 
2x(2x4)+6x 44+442+4+2)=2. 
6. Explain the figures at the left. 
4 Find the entire surface of prisms — 
7. 10 in. long, 6 in. wide, 4 in. 
; thick. 
§.=12 it, long, 9 ft. wide, 6 ft. 
high. 
9. 20 ft. long, 14 ft. wide, 10 ft. 
high. 
10. 16 by 18 by 4; 20 by 1 by 1. 
11. 12 by 9 by 8; 23 by + by 16. 
12. 12 by 12 by 6; 2 by 3 by 73. 
249. — To Find the 1. Mention several common objects that 
Contents of a are perfect cylinders; that is, of uniform 
Cylinder. diameter, and with ends (or bases) that are 


rite equal parallel circles. 

9. How might a cylinder be turned from 
a square prism of the same diameter ? 3. Recalling the formula 
for the area of a circle, 0.7854 of D? (p. 129), what part of the prism 
would be shavings, and what part cylinder ? 


138 MEASUREMENTS. 


4. Give the contents of the largest cylinder that may be turned 
out of a square prism 25 in. long, 4 in. wide. 


Explain the statement : — 4’ x 25 = « = contents of _. 
0.7854 of « = y = contents of 


5. A circle is ——. aan of a square of equal diameter. 


A cylinder is ——— of a square prism of equal diameter and 
length. 

6. Find the contents of a cylinder 10 ft. long and 2 ft. in 
diameter. 

Explain the statement : — 0.7854 of (2? x 10)= @ cu. ft. 

7. A cylindrical pail 6 inches in diameter inside and 12 inches 
deep contains # cubic inches. Forecast the result, observing that 
0.7854 is a little more than 3; thus, 3 of 6? x 12 = 324+. 

8. A cylindrical tank is 10 ft. deep and 8 ft. in diameter. 

9. A well is 32 ft. deep and 5 ft. in diameter. 

10. A gallon contains 231 cu. in. To hold a gallon, a pail measur- 
ing 33 sq. in. on the bottom must be w inches deep. 


10000 


* 


250.—To Find 1. In form, the ends of a cylinder are 
the Surface equal ..s. The rest of the surface is the 
of a Cylinder. convex surface. 


2. Suppose the diameter of a cylinder 
to be 4 inches; its circumference = 2, or 
3.1416 x D (§ 235). 


Written. 


OF CYLINDERS. 139 


3. The circumference of a cylinder is 
8 inches; its diameter is a, or 0.31831 x C. 

4. A cylinder is 20 inches long and 4 
inches in diameter. Find the area of its 
ends. Explain the statement :— 


(0.7854 of 47) x 2= 0. 
5. Roll an oblong paper to form a cyl- 


inder. Give the length and circumference 
of the cylinder thus made. 


20 


6. Unroll the paper and give the di- 
mensions of a rectangle equivalent to the 
convex surface of the cylinder. Explain 
the diagram at the left. 

7. The convex surface of a cylinder 
=Cx wl. Explain. [Z =length.] 

8. A cylinder is 25 inches long, 4 inches in diameter. Its con- 
vex surface is % Explain: (3.1416 x 4) x 25=2. 

9. A cylinder is 20 inches long, 5 inches in diameter. Entire 
surface ? 

10. If we allow 17 square inches for seams and the flange of the 
cover, how many square inches of tin are actually used in making a 
coffee can 6 inches in diameter and 8 inches deep? Show why there 
must be an allowance for waste. 


251. Oral Review. 1. A bookcase has 10 feet of space right 
and left and 6 feet up and down. It is 10 inches or “ feet deep, and 
y 


the cubical.contents = z. 


2. The sides that make the right angle of a triangle are each 10 
feet. The area is a. 


3. Give approximately the area of the surface of a lead pencil 4 
inch in diameter and 8 inches long. 


4. An old tree is 22 feet round; how far is it through ? 


140 MEASUREMENTS. 


6. Which takes more room, a cord of wood ora 5-foot cube ? 
6. 4a circle = a rectangle having the radius for one side and 
for the other. 
7. A 20-foot log averages 1 square foot in the cross section. The 
cubic contents are a. 
8. How many cubic yards of earth will a bin hold that is 
patts XPLOUE. xe, 
9. About how many cubic yards does your schoolroom contain ? 
10. What is the approximate capacity of a well 40 feet in depth 
and 7 feet in area of opening ? 


252. — Review 1. Give the dimensions of three dissimilar 
Problems. rectangles each containing 56 square inches. 
Written. Give the perimeter of each. 


2. A square rod contains # square feet. 
The wall of a rectangular cellar encloses 2 square rods. One of its 
dimensions is 20 feet, the other a. 

8. The boards of an old floor are 18, 14, 12, 10, and 6 inches 
wide. If used in equal proportion, what is the average width? To 
cover 2 squares, how many running feet would be required? 

4. A panelled ceiling contains 72 squares 14 feet wide. It is 12 
feet on one side, # on the other. 

5. Divide the area of a square on the diameter of a circle by the 
area of the circle. The quotient is a. 

6. Explain the formula C=7 x2 R. 

7. A board 6 feet by 6 inches contains 324 cubic inches actual 
measure. How thick is it ? 

8. How many oranges 3 inches in diameter will go into a box 
2x1x1 feet if packed in equal rows ? 

9. On a scale of 1 inch to 1 mile, represent a tract of land 2 
miles by 3 miles. Divide into square miles by dotted lines. Draw 
a mile square in the middle, and divide the rest into 4 equal tracts.. 

10. Each of the four contains @ acres. 


MISSING FACTOR FOUND. 141 


253. — To find a IP ee umes. Oe se Tex 1 98: 
Missing Factor. 2, Multipheand = 25; product = 400. 
Written. How is the multiplier found ? 


& 186+ a%='31. 
4, Dividend and quotient being given, how is the divisor found ? 


5, When product and multipher are given, how is the multiph- 
cand found ? 


6. What is the area of a rectangle 12 feet long and 614 feet wide ? 

7. A rectangle containing 108 square inches is 9 inches wide. 
How long is it? (9 x @ sq. in. = 108 sq. in.) 

8, A lot of land is 200 feet long and contains 24,000 square feet. 
How wide is it ? 

9. A sidewalk 50 feet long requires 50 square yards of concrete. 
How wide is the walk ? 


10. One-half an acre of land is taken for a new street 40 feet 
wide. How long is the street ? 


ll. The area of a triangle is 325 square inches; its base is 25 
inches. What is its altitude ? 


12. The altitude of an isosceles triangle is 14 feet; its area is 126 
square feet. What is its base ? 

18, At 30¢ a board foot a mahogany board one inch thick and 12 
feet long cost $2.70. How wide was it? 

14, The area of a rhomboidal field is 12 acres. Its length being 
20 rods, what is its altitude ? 

15, A square contains 400 square inches. How long is it? 

16. The perimeter of a square is 1000 feet. Its area? 


17. The radius of a circle is 5 feet; its area is 300 square feet. 
What is the circumference? (}C x R= A.) 

18. What is the area of a circle 100 feet in diameter? 
oie 0.7854 — A) 

19, What is the diameter of a circle containing 7854 square feet ? 
(2 x 0.7854 = 7854 sq. ft.) 

20. A circle contains 28.2744 square inches. What is its diameter ? 


142 MEASUREMENTS. : 


254.— Contents and l1138xT7x$a= §$ 910. 
Two Dimensions of a 2. I hired 15 men at $2.50 per day each. 
Solid given, to find the At the completion of the work I paid them 

Third Dimension. in all $150. How many days did they 

work ? 

3. A box on my table holds 432 cubic inches. It covers 72 square 
inches of the surface of the table. How high is the box ? 

4. The area of the floor of your schoolroom is 900 square feet. 
The room contains 10,800 cubic feet. How far is the ceiling from 
the floor ? 

5. A packing box is 48 inches long and 30 inches wide. How 
deep must it be to hold 10 cubic feet ? 


10 x 1728 
Statement : SRE ane a depth. 
Explain the statement, and show a short solution. 
6. A closet 8 feet high and 27 inches deep will contain 72 cubic 
feet. How wide is it? 
7. A pile of 198 cords of 4-foot wood covers 16 square rods. How 
long is it? How high is it ? 
Explain the statements : — 
— 198 x 128 _ 
16 x 2724 
8. A cylindrical oil-tank holds 10 gallons. Standing on the 
floor it covers 77 square inches. How high must it be ? 


A eed om B 


9. A bookcase holding 32 cubic feet covers a wall space of 24 
sq. ft. How far must it project into the room ? 


10. I have room in my stable for a grain bin 8 ft. by 4 ft. How 
deep shall I make it to have it hold 72 bushels ? 

11. A grindstone 4 ft. in diameter contains 6.2832 cu. ft. How 
thick is it? Explain the statement: 6.2832 + (4° x 0.7854) = a. 


12. In digging a trench 3 ft. wide and 43 ft. deep 330 cu. yds. of 
earth were removed. How long was the trench ? 


PROBLEMS. 143 


255. — Miscellane- 1. I buy a corner lot 120 ft. by 50 ft. and 
ous Problems. use the earth obtained by digging a cellar 60 
Written. ft. by 30 ft. by 10 ft. to raise the grade how 

many feet ? 

2. A circular standpipe 75 ft. high is 25 ft. in diameter. When 
2 full, how many gallons of water does it contain, reckoning 7} 
gallons to a cubic foot ? 

3. A speculator buys a field 600 ft. long and 500 ft. wide for 

2500. He runs a 40-ft. street through the centre in each direction 
at an expense of $425 for labor. He sells the land at 20 cents a 
square foot. How much does he make or lose ? 

4. At $3.75 per square yard what will be the cost of paving 3 of 
a mile of street 81 ft. wide ? 

5. A reservoir supphes a town with 4,575,800 gallons of water 
daily. If its surface area is 7 acres, how much will the water be 
lowered in a week, provided one-half as much runs in as runs out ? 
Call 1 cu. ft. equal to 74 gal. 

6. In a house of 36 windows a glazier finishes drawing the sash 
of 4 windows in 3 h., spending twice as long on the inside as on the 
outside. He can do the outside of them all in 1 day. How long is 
a day’s work ? 

7. A water glass has two bands round it, each containing ten 
figures. It takes three seconds to cut each figure. What will it cost 
to decorate 2,1, gross at four dollars an 8-hour day ? 


8. Find 80% of as many articles as can be bought for $200 at 
162¢ each. If sold ata profit of 100%, how many would be sold 
for $2? 

9. What will settle a debt of $127.50 that has been drawing 9% 
interest for 248 days ? 

121 x $36 x 162 x 75 
10. 183 gg ame a oe: SEAT | 
if © 81 x 374 x 374 z 
1l. From a lot of land 40 rods square I sold 40 square rods. 
What is the remainder worth at $230 an acre ? 


144 


MEASUREMENTS. 


256. — EXAMPLES FOR PRACTICE. 


For dictation. 
1. Give the perimeter of a 6- 
inch square. 
2. Of 4 
3. What is the ratio of the 


diameter of a circle to its circum- 
ference ? 


a 6-inch square. 


4. 8 feet 3 inches is what part 
of a rod? 

5. How many square feet in 
4 a square rod ? 

6. T'wo angles of a triangle 
measure 30° each. What -~does 
the third angle measure? Of 
what kind is the triangle ? 

7. How many cubic inches in 
a cube 2 of a foot long ? 

8. Find the entire surface of 
a 6-inch cube. 

9. Of 3 a 6-nch cube. 

10. 4 cord feet cost $ 5. 
will 3 cords cost ? 

11. Your schoolroom is 12 
feet high and contains 10,800 
cubic feet. Length of floor ? 


12. Contents of a 4-inch cube ? 


What 


13. Of one twice as long? 


14. Rods in 31 miles ? 


15. Length of a square con- | 
a square mile ? 


taining 900 square feet ? 


At sight. 
1. 0.0001 of 24,765 = a. 
2. 2 of 1 rod = @ feet> 
3. Days from Oct. 17 to Dee. 
25, inclusive. 
4. Average temperature for a 


week, if the thermometer read: 
7°, —4°, 10°, — 6°, —18°, 12°, 
20°? 


5. Area of circle 100 feet in 
diameter ? 

6. Cost of 1 pound if # pound 
cost $2? 

7. What is the exact middle 
of February, 1900 ? 

8. Area of rhomboid when 
base and altitude are 24 inches ? 

9. Area of square 163 feet 
long? 

10. Number of board feet in a 
board 12 feet long, 8 inches wide 
at one end and 10 inches at the 
other ? 


ll. V1i44 — V81i =2. 

9 25 x 6} x 38 _ 

19x13 x -V25 | 
18. 9 yards @ 137¢ cost = a. 
14. Area of triangle 61 by 3. 
15. How many acres is 61% of 


PROBLEMS, 145 


257. — Practical 1. 6}, 8, 44, are the dimensions of my 
Exercises in Mensu- coal bin. Reckoning 90 pounds to a cubic 
ration, etc. foot, what will a hin full cost @ $5? 

Written. 2. Quincy granite weighs 1653 pounds to 


the cubje foot. What is the weight of 6 pieces of curbing 8 inches 
thick, 2 feet wide, and half a rod long ? 

3. Find the cost of carpeting a 9-foot hallway 22 feet long with 
three-quarter carpeting at $0.87}. Cut no strip, and allow 11 feet 
per strip for matching. 

4. How many tons of 15-inch ice may be cut to the acre, a cubic 
foot weighing 57} pounds? Apply your knowledge of cancellation. 

5. What is the capacity, in 42-gallon barrels, of a cylindrical oil- 
tank 31 feet in diameter, 22 feet long? Make a statement and 
cancel. 

6. What is the area of a sector of 120°, its radius being 24 inches? 

7. A ball ground 375 feet long and 280 wide is enclosed by a 
tight board fence 8 feet high. What will the boards cost at $24 
per M.? Add 10% for waste. 

8. Bought 12,000 long tons of coal at $4.00 and sold it at the 
same price per short ton. What did I gain? 

9. What will it cost to polish the visible portions of a shaft of 
red granite 6 feet by 2 feet by 22 inches at 62¢ per square inch ? 

10. Draw a6-inch square, a rectangle 9 inches by 4 inches, and one 
} inches by 12 inches. Compare areas and perimeters. “What infer- 
ence do you draw ? 


258.— Examples for 1. What decimal of a square prism becomes 
Practice. shavings when the largest possible cylinder is 
Written. turned from it ? 
2. What number subtracted 88 times from 80,005 will leave 13 
as a remainder ? 
3. A railroad company fences 13 miles of its road at 732 cents a 
rod. 
4. How many square feet of zinc will line a cubical cistern 5 ft. 
10 in. deep ? 


146 MEASUREMENTS. 


5. The time of the operatives in a mill was increased from 52 to 
58 hours, and their wages increased ;45. Was this a gain or a loss to 
them ? 

6. Bread sells for 10 cents with flour at $5.00. Flour goes up 
to $6.50. What should bread sell for on this basis ? 

7. Ina city of 7200 school children there are 2720 cases of tardi- 
ness in a year during which there are 400 sessions of the schools. 
The average attendance is 6800. How often is each child tardy ? 

8. Find the cost of six 8 x 10 sills 18 ft. long at $24.75 per M. 

9. In a school containing 567 white children every tenth child is 
colored. How many children in the school ? 

10. A schoolroom measuring 32 ft. x 284 ft. x 13 ft. seats 49 
pupils. Each one needs 1800 cu. ft. of fresh air an hour. The 
room full would last the class « minutes. 


259.— Problems for 1. How many sheets of paper folded into 
Analysis. 16 leaves will make a, 400-page book ? 

Oral. 2. At $10.50 a week what is the amount 
of your board bill from noon of Aug. 21 to 
noon of Sept. 25 ? 

38. What is a £100 Bank of England note worth in New York at 
its face value ? 

4. Cost of 82 yds. at $ 0.374 per yard. 

5. Compare a 5-inch square with one half as long. 

6. A circle is 10 feet in diameter. How long is an arc of 36° 
in its circumference ? 

7. A cubic foot of distilled water at a temperature of 38° F. 
weighs 1000 ounces. How will you find the weight of a gallon ? 

8. # of an acre produces a crop that sells for $360. How much 
is this for every 12 sq. rds. ? 

9. I pay $1.80 for having my cord-wood sawed into 3 sticks. 
What ought I pay when it is sawed into 4 sticks ? 

10. A trapezoid is twice as wide at one end as at the other. It 

measures 12 in. in the middle, x in. at one end, and y in. at the 
other. 


DEFINITIONS. 


147 


260. DEFINITIONS. 


[FOR REFERENCE. ] 


Acute Angle. 
than a right angle. 

Altitude. Height. Measured by a 
straight line perpendicular to the line 
of the base, and extending from it to 
the highest point. 

Angle. The divergence from a com- 
mon point of two lines having different 
directions. 


An angle sharper 


Are. Any portion of a circum- 
ference, 

Area. The size or total contents of 
a surface. 

Base. ‘The line or surface on which 


a figure is supposed to stand. 

Chord. A straight line joining the 
ends of an arc. 

Circle. 
by a curve every point of which is 
equally distant from a point within 
called the centre. 

Circumference. 
boundary of a circle. 

Convex Surface. The surface of 
a solid excluding that of its bases. 

Cube. A solid with six square 
faces. 

Curvilinear surfaces 
bounded by curves. 

Cylinder. A solid having for its 
bases equal parallel circles, and hay- 
ing a uniform diameter. 

Degree. <A 360th part of a circum- 
ference. 

Diagonal ofa Polygon. A straight 
line connecting the vertices of two 
angles not adjacent. 


The perimeter or 


are those 


A plane surface bounded | 


' shortest distance across a circle 


Diameter. <A line measuring the 

or 

square through the centre. 
Dimensions. Measurements needed 


to find contents. 


Equiangular. Having equal angles. 
Equilateral. Having equal sides. 
Figure. A surface bounded by 


lines or a space bounded by sur- 
faces. 
Horizontal. 
of the horizon. 
Hypotenuse. The longest side of 
a right triangle. 
Inclined. Neither horizontal nor 


Parallel to the plane 


| vertical. 
Isosceles triangles have two sides 
equal, 
Line. The limit of a surface. The 


path of a point. 

Oblique lines are neither horizon- 
tal nor vertical. Oblique angles are 
greater or less than right angles. 

Oblong. A rectangle whose length 
exceeds its breadth. 

Obtuse angles are greater than right 
angles. 

Parallel. Extending in the same 
direction, and in all parts equally dis- 
tant. 

Parallelogram. A quadrilateral 
whose opposite sides are parallel. 

Perimeter. The circumference of 
a surface or the sum of its bounding 
lines. 

Perpendicular. At right angles to 
another line or surface. 


148 

Plane. A plane surface is a flat or | 
level surface. 

Point. That which has position, 


but no length, breadth, or thickness. 
The end of a line. 

Polygon. 
straight sides, commonly more than 
four. ° 

Prism. A solid whose sides are 
parallelograms, and whose bases are 
equal parallel polygons. Prisms are 
named from the form of their bases, as 
square prisms, rectangular prisms, tri- 
angular prisms, hexagonal prisms, etc. 

Quadrant. A fourth part of a cir- 
cle or of a circumference. 

Quadrilateral. A plane surface 
having four straight sides. 

Radius. A straight line extending 
from centre to circumference of a 
circle. 


Rectangle. A parallelogram hay- 
ing four right angles. 

Rectilinear. Bounded by straight 
lines. 


Rhomboid. A parallelogram with 
oblique angles. 
Rhombus. 
boid. 

Right Angle. An angle of 90°. 

Scalene triangles have their sides 
unequal. 

Secant. A straight line that cuts a 
curve at two points. 

Sector. The part of acircle bounded 
by an arc and two radii. 


An equilateral rhom- 


A plane surface having | 


MEASUREMENTS. 


Segment. The part of a circle be- 
tween an arc and its chord. 

Semicircle. Half of a circle. 

Sextant. One-sixth of a circle. 

Solid. A form having three di- 
mensions, — length, breadth, and thick- 
ness. 

Square. An equilateral rectangle. 
A plane surface with four equal sides 
and angles. 

Surface. That which has only two 
dimensions, — length and_ breadth. 
The outside of a solid. 

Tangent. A line touching a curve 
at a single point without crossing. 

Trapezium. <A quadrilateral no 
two of whose sides are parallel. 

Trapezoid. <A quadrilateral only 
two of whose sides are parallel. 

Triangle. A plane surface having 
three straight sides. <A right triangle 
has one right angle ; an obtuse triangle 
has one obtuse angle ; an acute triangle 
has three acute angles. 

Vertical. Relating to the vertex. 

Vertical lines point towards the 
zenith and the earth’s centre. 

Vertex. The point in an angle 
where the sides meet. 

a. <A Greek letter pronounced like 
p. It stands for 3.1416—, the ratio of 
the circumference of a circle to its 
diameter. 

Oo, /’, , Sometimes used for feet, 
inches, and lines or twelfths of an 
inch. Compare their use on p. 9. 


PERCENTAGE. 149 


Computing by Hundredths. 


261. — Percentage: One hundred is the common standard of 
Hundredths of a comparison. The merchant gains 10 on 
Number. every 100, or 10 per cent. The rate of 
Oral. interest is 6%, or 6 on every 100 used. The 


commission paid is 2 on every LOO spent, 


or 2 per cent. 80% of the lquor is alcohol, that is 30 parts in 


every LOO. 


6. 
ff 
8. 
9. 


1. What is meant by saying: 


12% of the grain spoiled ? 
334 per cent of the month was stormy ? 
14% of the scholars were absent ? 


Give other illustrations. 
2. What is the meaning of per cent, the short form of per centum ? 


3. 25 per cent of 200 tons is 50 tons. 


50 tons is w per cent of 200 tons. 
50 tons is 25 per cent of y tons. 
zis 25% of 200 tons. 


The rate per cent, or the number of hundredths, is a. 
The base, or the number a part of which is to be found, is y. 


The percentage, or part of the base required, 1s z. 
4. The following expressions are alike in value: 
| 25 per cent = 25% = 25, = 0.25 = 4. 
Which are fractions ? Decimals? Which is most easily used ? 
5. Of 72 find 50%; 4+; 75%; 4; 162%; 334% ; 662%. 


tof $48 =2 10. Base, $60; rate %, 25; percentage, x 
121% of 96 lb. =a 11. Rate, 75%; base, 400 yr.; 300 yr.= 
20% of 90 =a 12. 331% of his time, or x hours, he sleeps. 


50% of 2000 = x. 13. The loss was 624% of $1000, or a 


150 PERCENTAGE. 


262. — The Whole 1. A farm that cost $5400 was sold for 
Given: a Part of it 75% of its value. What was the selling 


to be Found. price ? 

5 ay aay 2 2. Show that 75% = 3. 

3. Of the methods, A and B, which seems 
iy preferable? Why ? B. 

4 1350 4. In general, in multiply- $ 5400 
75 th or 7 of $ $499 = $4050. ing the base by the rate per 0.75 
| cent to find the percentage, A a 
will it be better to use the © 4060.00 


decimal form as in B, or the 

equivalent common fraction as in A’ 

6. What part is $ 4050 of $5400? What % ? 

6. 9is of what? $4050 is 3 of a $4050 is 75% of a. 

The Gh the $ 27,000 paid for an Bits 121% was in cash, and the 
remainder in notes. What was the cash payment ? 

8. Of 12,650 bushels of grain, 34% was in corn, 28% in oats, and 
the remainder in wheat. There were a bu. of corn, y bu. of oats, 
and z bu. of wheat. Explain the statement: 


[100 % — (34% +28%)] x12650 =z. 


How much is— Find a discount of — 
9. 25% of 3742 tons? 12. 15% on 63 yards @ $2.50. 
10. 71% of 784 miles ? 18. 374% on a $558 piano. 
ll. 162% of 5733 acres ? 14. 182% on 42 tons (@ $ 6.59. 


15. Compare $ of $400 with #% of it. 

: acd : OO S280 25 bie ylain: ae nn ge Te 
16. Read: 0.002; 2%; 160 Explain 100 x 6 600 |p 
17. What is 7 of $64,000 ? 

18. Find 7% of 64,000. 
19. My property is assessed for $ 24,800. Tax rate 12%. My tax? 


20. A city increases 24% in 10 years; that is from 37,860 popula- 
tion to a. 


TO FIND THE BASE. 151 


263.— A Part Given: 1. } of my age is 16 years. How old am I? 
the Whole to be Found. 2. 5()% of my money is $80. I have $2. 


Oral. 3. $ of the price was $36; 4 of it was 4 
Bow. of $36, or $x, and 7 or the whole of it was 
R 


7x $avor $y. 

4. 6% of my salary was $72; 1% of it was 4 of $72, or $a, and 
100%, or the whole, was 100 x $a, or $y. 

5. The percentage, $18, is 75%, or 8, of the base, which must be 
$a, for 18 is 3 of z. 

6. There are 36 cups with saucers in a set of crockery. This is 
334% of the set. There are x pieces in the set. 

7. Can you find the whole of a number when you know that 75% 
of it is 150? 

8. Base x Rate % = Percentage. When the product and one 
factor are given, how is the other factor found ? 

9. 2 x 40 = 25; hence 25 + 3=2; and 25+ 40=y. 

10. 12% of 800 = 96; hence 96 + 12%, or 96 + 0.12 = a. 


264.—A Part is 1. Compare 2 and 4: 2 is dof 4 or 50% 
what Fraction of the of 4. 41s 2 times 2 or 200% of it. 


Whole? 2. What part is 3 of 6? what % of it? 
Oral. 8. 5 is what part of 20? what % of it? 
Ros 4. 16 is one . or x% of 48. 


5. $24 is two . or x% of $36. 

6. 12 ounces is x fourths of .a pound, or ¥% of it. 

7. 18 ewt. is .. of a ton, or 7% of it. 

8. 1% of 2000 lb. is a lb: 800 lb. is as many per cent of it as 
« lb. is contained times in 800 lb., or 7%. 

9. Base x Rate % = Percentage. The product is 48, one factor 
is 3, the other is a The product is the percentage, one factor is the 
base, the other factor .., for R= P + B. 

10. 16=27% of 40. ll. 48=2% of 64. 12. 96=2% of 144. 


152 PERCENTAGE. 


265.— To Find 1. Base = $3125, rate % = 48, percentage 
Percentage, Base,or = $1500. With any two find the other, and 
Rate %. explain the operation. 
Written. 
To find P. To find B. To find R. 
$ 3125 $ 3125. 0.48 
0.48 ,48.)B 1500, 00. 3125)$ 1500.00 
25000 144 12500 
12500 60 25000 
% 1500.00 48 _ 25000 
120 


2.23% of a cargo of 96 
coal was thrown overboard 9240 
to save the schooner. She PA() 
sailed with 4320 tons, reach- 
ing port with a. 

38. 72% or 4536 volumes in a library are not works of fiction. 
There are 2 volumes in the brary and y volumes of fiction. 

4. 6035 persons bought tickets to a fair. This was «% of the 
8500 that attended. 

5. 625 scholars belong in the Lincoln school: 600 of them are 
present, or # % of the whole. 

6. 57 % or 11,100 tons of an ice crop remained unsold. There 
must have been w tons in the whole crop. 

7. The cargo of the Sea Hing was valued at $38,475. The 
value of the cotton was 162% of the whole, that of the sugar 374%. 
The miscellaneous part of the cargo was valued at @ dollars, or y % 
of the whole. ‘Take the shortest method. 

8. I sold my bicycle for $85. It cost me $125. I must have 
lost x % of the cost. 

9. If I had lost but 15 %, I should have sold it for x $. 

10. 192 pages of a book of 432 pages are illustrated. ‘This is 
« % of the whole. 


TO FIND ANY TERM. 153 


266. — To Find 1. Which is most profitable, a gain of 3 
Any Term when the per dozen, 5 per score, 25 per cent, or 36 per 
other Two are Given. gross? Why? 

Oral and Written. 2. Compare 2 of something with 2% of it. 

3. Six wrong out of 24 problems solved is 
# wrong out of a hundred, or 7%. 


lool 


4. The Brooklyns won 7 games in their series with Boston, the 
Bostons won 4, and one game was a tie. The winners’ per cent 
was &. 

5. Thirty-six hits in 80 times at the bat is a batting average 
of «%. 

6. The centre fielder has 80 chances, and makes 4 errors. His 
fielding average is 7%. 

7. The crew pulled 56 strokes to the minute at starting, but fell 
off to 30 at the finish. This was a loss of what per cent ? 


8. 9. 10. 
2% of 600 =x 2% of 800 =x w= 871% of 128 
4% of “v2=2 8% of w=12 “% of 144 = 120 
“% of 1200 = 8 “% of 200 = 4 83£% bad and w% good 
11. 12. 18. 
25% of 53? 19 is 7% 57 25 is 4% of x 
30% of 400? 70 is e% 2100 280 is 14% of x 
81% of 22? 162 is 7% 662 Sis 4% of a 


14. I paid 2% commission to my agent for selling a farm for 
$1250. How much money did he have left to send me ? 


15. 21% was paid a collector who earned $22.50 a month in this 
way. What were his annual collections ? 


16. Of a farm of 320 acres 108 acres are given to wheat, 96 acres 
to oats, and the remainder to corn. What per cent of the farm are 
the cornfields ? 


154 


Find the value of a. 


PERCENTAGE. 


267.— For PRACTICE. 


Oral. Written. 
Rate % Base. Percentage. Percentage. Base Rate % 
1 | 162 $ 9.30 x 1 | 32), m. 2674 m x 
2 | 124 x 123 tons 2 170.40 | x | 173 
sh he 7) yd. 18? yd. 3 | 3618 yr. 130 yr wx 
~4 | 374 x 57 4 181 A. x 453 
5 | 40 | 9000 m. x 5 | 4857.6 ft. | 5280 ft. x 
6 x $ 0.50 $ 0.314 6} $5.76 x 4 
7| 2 v 14 d. T | ‘$5400 $ 9600 x 
8 | 874 | 13 tons aw 8 | ay 1500 ed. | 15 
ey 160 1062 9 | 204 sq. ft. | 5200 sq.ft.) a 
10 | 61 z 15sq.m. | 10 | 6500 T. x 83} 
LIT Do 725 x ll 143 d. 365 d. x 
12 2 | a century 8 mo. 12 |$13,651.56| $75,842 x 
13 | 90 608 bu. x 13 a 36 x 
14 | 182 wv 18 bales | 14 84, x Zt 
15 | « | To reams | 61 reams | 15 ay 
16 8 $ 12,000 x 16 3288. x 94 
17 | 314 x 30 cords | 17 | $349.06 $ 9006 x 
18 | « 726 gal. 605 18 $18 x 2 
19 & | $120,000 x 19 x $8100 5 
20; ¢ ar Od Ib. 20 a 3 3 


268. — Profit and 


Loss. Problems. 


1. Sold a house that cost $5000 at a profit 


of 30%. 


Proceeds of sale ? 


2. Amerchant’s sales for January amounted 
to $28,000, but 12% was lost in bad debts. 
The net proceeds of the sales for the month were w dollars. 


Written. 


PROFIT AND LOSS. 155 

% 
3. Gained $12 or 20% in selling a Century Dictionary. It cost 
me @ dollars, and I sold it for y dollars. 


4. A sewing machine cost me $24. I sold Gain or loss is 
it for $32. I gained x%. always a part of 


the cost. 


4 32 — 24 
Explain the statement : RTE Ta xp. 


5. A conductor’s wages were $72 a month. They were reduced 
to $60. This was a cut-down of 7%. 
72 — 60 
oe : 
| 72 70 
6. Cost, $8000; selling price, $6000; loss per cent, 2. 
7. Cost, $6000; selling price, $8000; gain per cent, 2. 


8. Which is more profitable, to buy cloth for $3 and sell it for 
$ 3.50, or to buy it for $4 and sell it for $4.80 ? 


9. Gas is reduced from $2 to $1.60 per 1000. How much do I 
save on $45 worth of gas? 


Explain the equation: 


10. Last winter my coal cost me $6 a ton. This winter I pay 
$6.50. This is an increase of what per cent? 


269.— Amount and 1. Bought wood at $4 a cord and sold at 
Difference. a gain of 20%. What did I sell it for ? 


Written. a. $4-+20% of $4=2. 
b. 120% of $4=-2a. 
Explain the two statements. Which suggests the shorter solution ? 
2. Sold a typewriting machine that cost me $80 at a loss of 
10%. What did I receive for it ? 
a. $80 —10% of $80=2. b. 90% of $80 =2. 
Explain the two statements. 
3. Any number is 2 fifths, y tenths, z hundredths, w% of itself. 


The Base added to the Percentage is the Amount. 
The Base less the Percentage is the Difference. 


156 PERCENTAGE. 


4. Sold a dwelling-house for $7500 at a profit of 25%. It cost 
me # dollars. 
5. An epidemic decimated a southern village, leaving it with but 
639 inhabitants. How many died ? 
6. A farmer who owned 390 acres had increased his farm 30% 
within two years. How much did he own at first ? 
7. A speculator lost $3000 or 6% of his property. What was 
it then worth ? 
8. I sold a yacht for $800 at a loss of 60%. Required its cost. 
9. In a certain class 15% sing soprano, 45% sing tenor, 30% 
sing alto, and 6 sing bass. How many are there in the class ? 
10. A piano that cost $450 was sold for $292.50. What was the 
per cent of loss ? 


270. — Problems. 1. What is 2% of 250 ? 
For dictation. 2. 50% of a man’s age is 15 years. How 
old is he? 
What per cent of the day has passed at 9 A.M. ? 
Cost, $2; gain %, 16; selling price ? 
Selling price, $6; gain, 20% ; cost? 
Gain, $3, or 5%; cost? Selling price ? 
Selling price, $18; cost, $15; gain % ? 
8. Every 16th coin out of a collection of 176 silver dollars was 
counterfeit. - This is a loss of what per cent ? 
9. 10 per dozen is what per cent? 
10. The square root of 625 is 10% of what? 


st eS Co 


271. — Problems. 1. In 33 what per cent is the numerator of 
At sight. the denominator? In 3? 
2. What is 32; of a rod in feet? In inches ? 
3. 231 cubic inches =1 gallon. Separate 231 into its prime 
factors. Give the dimensions of a tin pan that will hold a gallon. 
4. What part of a year is in the longest months ? 


PROBLEMS. 157 


5. What is +% of 21,000? 
6. What per cent of the rotation of the earth is accomplished at 
10 a.m. ? 


7. 334% of 60% of 4 of the money remained. How much did 
the thieves take ? 
8. My property is assessed for $2500. The rate of taxation is 
21%. What is my tax ? 
9. What per cent of the surface of a 4-inch cube is on five sides 
of it? - 
10. Bought thread at 4 cents a spool and gained 300%. It sold 
for # cents. 


272. — Problems. 1. 84% of a yard = x&% of a foot. 
At sight. 2. Three sides of a square=x% of its 
perimeter. 
3. w% of the day has passed at 9 p.m. 
4. Dig 4.75 3) of 64 is 25% of a. 
5. 16 is 2 of a and 2% of y. 
6. A ee is *% of a ream. 


7. Gave $24 to James, and $30 to Lucy. Lucy had #% more 
than James, and he had 7% less than Lucy. 

8. Paid the price of a pound for 14 ounces. I thus lost x%. 

9. In a series of ball games the Alphas won 40% and the 
Omegas 50%. Two games were drawn. How many were played? 


10. V9=2% of V144. 


273. — Problems. 1. To 20 gallons of alcohol add 5 gallons 
of water. What per cent of alcohol in a 
quart of the mixture ? 

2. Sold a mine for $72,000, and gained 20%. It cost me x dol- 
lars. $72,000 + (cost % + gain %) = cost. 

3. I purchased a patent for $8000. The seller lost 84% of its 
original yalue, which was « dollars, 


For analysis. 


158 PERCENTAGE. 


4. I paid $125 for what I thought was 4-foot wood. It proved 
to be but 45 inches long. What deduction should be made in the 
settlement ? 


5. Sold telephone stock for $25,000, at an advance of 25% on 
what I paid for it. What did I gain ? 


274. — Profit and 1. $12 or 121% (of cost) is gained; cost 
Loss. = $a. 

Oral or written. 2. $8 or 162% is lost; cost = $a. 

38. $24 = cost; 331% is gained; selling price = $ a. 

4. $35 = cost; 142% is lost; selling price = $ a. 

5. $36 = selling price, which includes the cost and a gain of 
20% of the cost. $36 =cost++4 of cost, or $ cost; $36 is $ 
of $a. | 

6. $28 = selling price, which is the cost less a loss of 20%. 
What part of the cost is the loss? The selling price is . of the 
cost; $28 is : of Dy. 


7. Bought a bicycle for $80, and sold it for $100. My gain 
per cent was @. 

8. If I had sold it for $60 I should have lost $y or x per cent 
of cost. 


9. An importer bought silk at $2.50 a yard and sold it to a 
retailer for $3, who sold it to the wearer for $3.50. What per cent 
of profit did each make ? 


10. Sold a watch for $119 and gained 162%. How much should 
I have gained or lost if I had sold it for $ 100 ? 


11. A thrifty clerk resolves to live on 60% of his salary. He 
spends $48 more than he intended, but still saves $300. What 
was his salary ? 


PROFIT AND LOSS. 159 


275.— Gain and 1. Which is more profitable, buying meat 
Loss. at 16¢ and selling at 19, or selling potatoes 
Written. at 64 ¢ that cost me 56 ¢? 


2. Butter sold at 28¢ yields no_ profit. 
What would be gained on $140 worth sold at 30 ¢? 

8. Milk bought at 20 a gallon is sold at 8¢ a quart. What per 
cent is gained if 25% of the quantity bought spoils ? 

4. Had I better buy my winter’s coal to-day at $4.00 a ton cash, 
or wait 6 months and pay $4.25? I can get 12% interest for my 
money. 

5. A 5% increase in wages means $200 more a month for the 
employer to pay. What was his annual pay roll before and after 
the increase ? 

6. Mr. H. earns $1200 a year selling carriages at 15% commis- 
sion, all expenses paid. The manufacturer makes a net profit of 
142%. If 50 carriages are sold, what is their average cost ? 

7. An unscrupulous dealer buys 50 gallons of alcohol and adds 
14 gallons of water, and sells the mixture at 10% below actual cost. 
What per cent does he gain? 

8. I am offered a 10% discount on a suit of clothes marked to 
sell at $60. I know that even then the dealer will make 123%. 
I offer $50 and get the suit. What per cent does the dealer gain ? 

9. I sell 2 of a lot of land at % the cost and get $200 for the 
remainder. The original cost being $1200, what is my per cent 
of loss ? 


10. What per cent is gained by selling coal at the rate of 4 of a 
ton for what 1000 pounds cost ? 

11. A farmer’s sheep cost him $200. One out of every seven 
dies, and he sells those that remain for $275. What was the gain 
per cent, the cost of keeping being $ 40 ? 

12. A merchant sold a stock of goods for $3042 and gained 17%. 
What % would he have gained or lost had he sold it for $2892 ? 

13. For what should he have sold it to gain 100 % ? 


160 PERCENTAGE, 


Interest. 
276.— Bankers’ 1. I pay 6 cents for a year’s use of a bor- 
Method, rowed dollar: what is the rate of interest ? 
Oat 2. What does the expression “6 per cent 


interest ”’ mean ? 


3. A year’s interest is what per cent of the principal? 4. What 
part of a year is 2 months? 2 months’ interest is #% of the princi- 
pal. 65. 20 months’ interest is y x #% of the principal, or 2% of it. 

6. 200 months’ interest is what per cent of the principal ? 


7. How is 1% of a number found? 10% of it? 100% of it. 


At 6% the interest of any principal 
for 12 months 6% of it; 
for  2months 1% of tt; 


for 20 months 10% of tt; 
for 200 months = 100% of it. 


To find the interest at 6% of $3880 for 2 yr. 7 mo. 


Process. 
Interest for 31 mo. of $880. 


Explanation of Process. 


G2 2 yr 7 m0; a mo. 


es ‘ 20 mo.= 38.00 
és 10 NO LOD 
2 otis, TNO, es 1.90 
“831 mo. = $58.90 10. 10 mo. is what part of 20 mo. ? 
How then is 10 mo. interest found ? 


. 20 mo. int.= ;, of P. How is 
this found ? 


11. How is 1 mo. interest found? The total interest ? 


12. Into what convenient parts would you separate the time, if 
you are to find the interest for 26 mo.? 37 mo.? dyr. 7 mo.? 
5 yr. 8 mo. ? 


INTEREST. 


277.— Bankers’ 


Method. 
Rate, 6%. 
Written. 
Interest for 47 mo. of $720. 
66 ‘ 90 mo. = 72.50 
¢ ‘¢ 90 mo.= £72.50 
he ‘¢ §&mo. =~ 18.125 
he oe oom6) = 7.25 


: 47 mo. = $170.375 


Find the interest 
3. Of $280. for 2 yr. 8 mo. 
4. Of $640. ford yr. 7 mo. 
5. Of $ for 4 yr. 11 mo. 
6. Of $ 73.50 for 1 yr. 4 mo. 


278.— The Time 
in 
Days. 
Rate, 6%. 
cipal. 


To find 75 days’ 
$72 at 6%. 


Process. 
Interest for 75d. of $72 
‘6 ‘ 60d.= 7.20 
66 ‘ 15d.= 1.80 
se ‘“ 75 d.= $9.00 


interest of 


{ft 


10. 


161 


Explain the following process of finding the 
interest at 6% — 
1. Of $725 for’3 yr. 11 mo. 


2. Of $278 for 1 yr. 7 mo. 


19mo. $278 
20 mo. = 27.80 
mor = 1.39 


19 mo. = $26.41 


What shall be paid for the use 
Of $649. 8 mo. 
Of $750. 4 mo. ? 
Of $ 295.75 for 5 yr. 11 mo. ? 
Of $641.86 for 35 yr. 


for 7 yr. 
for 8 yr. 


35 mo. 


1. How many days in an interest month ? 
2. In an interest year? 3. 
2 mo. or 60 days is wh of the principal ; for 
6 days it is {1 of 
To find 5755 0 


The interest for 

, 1 “= 
shy OF Goya OF the prin- 
of a number we 


At 6% the interest of any 
principal 
of it; 
sy Of tt. 


for 60 days = ty 
ie 


6 days =; 


100 


for 


4. 15 days is what part of 
60 days ? 


162 INTEREST. 


Explain the process of finding the interest at 6% — 


5. Of $196 for 115 days. 6. Of $119 for 89 days. 
Int. for 115 d. for $196. Int. for 89 d. for $119. 
60 d. = $1.96 60 d. = $1.19 
DU i ee OS 20d.= 0.8966 + 
20.-d.= 0.65383 + 6da;= 01190 
5d.= 0.16838 + 3d.= 0.0595 
115 a. = $3.7566 — 89 d. = $1.7651 _ 
What shall I pay at 6% for the use of— Find the interest at 6% of — 
1. $-780. for 67 d. ? 12. $94. for 200 d. 
8. $640. for 93 d. ? 13. $762. for 5 mo. 14 d. 
9. $92. for 3 mo. 12 d. ? 14. $815. for 86 d. 
10. $87.50 for 117 d. ? 15. $924. for 8 mo. 11 d. 
11 LOG stor Zane, Lid 2 16. $17.84 for 17 
279. — Bankers’ 1. 6% interest = $18; 1% interest = $a; 
Method. 5% interest = 5 x $a or Py. 
At any Rate. 2. 6% interest = $42; 7% interest = $a. 


3. What is 10% interest when 6% interest is $90 ? 
4. What is 4% interest when 6% interest is $72 ? 


5. Having found the interest on any sum of money at 6%, how 
shall we find it at1%? At any rate? 
To find the interest of $105 for 75 d. at 5%. 


Process. 6. Show how the interest 
6% int. for 75 d. of $105. at 6% is found. 
sige 7. At1%; at 5%. 
eat te Crane 8. aa the on had 
6)$ 1.3125 = 6% int. been 7%? 10%? 12%? 
0.2187 + = 1% int. 9. Find the interest of 
$ 1.0938 = 5% int. $ 280 for 72 d. at 8%. 


10. Which is easier, to find 8% interest by taking 8 times 1% 


interest, or by adding 2% interest to 6% interest? Try both ways 
before deciding. 


THE 


Find the interest of — 


ll. $640. 
12. $ 270. 
13. $382. 
14. $927. 
15. $ 864. 
16. $318. 
17. $725. 
18. $649. 


for 1 yr. 8 mo. at 1%. 


for 3 yr. 10 mo. at 14%. 


for 1 yr. 9 mo. at 2%. 
for 6 mo. 4 d. at 3%. 
for 2 mo. 7 d. at 4%. 
for 1 mo. 13 d. at 5%. 
for 29 d. at 7%. 

for 67 d. at TE%. 


19. $84. for 54 d. at 8%. 


20. $125. 


for 276 d. at 9%. 


21. $6000. for 118 d. at 10%. 
22. $1525. for 63 d. at 1%. 
23. $2500. for 93 d. at 21%. 


280.— The One Dollar 


Method. 


found first ? 


ONE DOLLAR METHOD. 


163 


(4 of 6%.) 
(4 of 6%.) 
(4 of 6%.) 
(4 of 6%.) 
(6% — 2%.) 
(6% —1%.) 
(6% +1%.) 
(6% + 14%.) 
(6% + 2%.) 
(6% + 3%.) 
(10 x 1%.) 
rz Of 6%.) 
[(k + ay) of 6%.] 


1. What two methods of computing in- 


terest have previously been presented ? 


2. In which one is the interest at 6% 


3. What is “a 6% method” of calculating interest ? 


4. In how many states is 6% the rate of interest established by 


law ?. 


[App., p. 15.] 


5. What is the interest of $1 fora year at6% ? 6. For 2 mos., or 


1 : raaor ?.. 
i of a year ! 
a cent of interest ? 


8. What is the interest of $1 for 6 days, 


7. At 6%, how long may I use a borrowed dollar for 


zy of 2 months ? ear: 


mill pays 6% interest on a dollar for how long? 


10. The interest of $1.00 
For 2years = $0. 
8 months = _ 0. 


For 
For 18 days 


For 2 yr. 8 mo. 18 d. = $0. 

11. Compared with the interest 
of $1, what will be the interest of $ 
Of $84.75? 


Of $ 725? 


md), 


[p. 20. ] 


For 2 months = 


For 6 days = 


7~ oO 


ios 


Bah est fe 
Of a dollars ? 


At 6%, the interest of $ 1.00 
For I year 


= $ 0.06 
0.01 
0.001 


Of $420? 


164 


12. The interest of $1.00 


FOroyt. == > 0: 

For 10 mo.= 0. 

For 24d... = 0. 

Total = $ 0. 

4. *Ehe int, Of 1 15. 
Hor (3 Yio = pV: 
Hor Moses" 47 
HOralp dee =: 
Tobabe= 90). 


281.— Interest by 
the One Dollar 
Method. 

At Any Rate. 


i 


Process. 


of $1 at 6%. 
— $0.18 
0.035 
0.0031 


$ 0.2182 
48.96 


816 
39168 
4896 

9792 

6)10.68144 

1.7802 at 1% 


$12.46 at 7% 
48.96 = Prin. 


Interest 
For 3 yr. 

LST fae va} 

re LO Cs 


it; 


III. 


IV. $61.42 = Amount. 


INTEREST. 


18. The interest of $1.00 


For. 4 yr. = $0. 
Hor, dm 0: 
Hor-10.0 See. 

Total = $ 0. 

The int. of $1. 16. The int. of $1. 
OL) <2 ta 0), For S547. = $0) 
For’-9 mo, = 770. Hor. omg. see 
For 22d. = 0. Hor desea 

Total = 0. Total = $0. 


I hire $48.96 at 7% for 3 yr. 7 mo. 19 d. 
What shall I pay at settlement ? 


Explain these four steps of the process: 
I. Finding the interest of $1 at 6%. 
II. Finding the interest of the given 
principal at 6%. 
III. Finding the interest at the given 
rate. 
IV. Finding the amount. 


2. In II. why is the smaller number 
used as a multipher ? 


Norr.—The work should be carried to four decimal 
places, and results given to the nearest cent. 


38. What will discharge a debt of 
$475 which has been drawing 5% inter- 
est for 2 yr. 11 mo. 24 d.? 

4. Find the amount of $7000 at 4% 
for 3 yr. 3 mo. 18 d. 

5. I hold two notes of $731 each, 
one at 5% interest, the other at 8%. 
They have been running 4 yr. 8 mo. 


17d. What shall I receive at settlement ? 


TIME BETWEEN DATES. 165 


Find the amount under the following conditions : 
6. Principal, $84.75; rate of interest, 4%; time, 3 yr. 15 d. 
7. Principal, $942; rate of interest, 5%; time, 4 yr. 1 mo. 7 d. 
8. Principal, $193; rate, 7%; time, 18 mo, 27 d. 
9. Principal, $64.50; rate, 8%; time, 5 yr. 5 mo. 5 d. 
10. Principal, $712.10; rate, 9%; time, 7 yr. 4 mo. 29 d. 
ll. 4 yr. 6 mo. 21 d., $425.50, 3%. 


282.— Time between 1. To tind the time in years, months, and 
Dates. days from June 24, 1895, to Aug. 18, 1896. 
Explain each process. 
Process A. Process B. 
From June 24, 95, to June 24,96 = 3 yr., or 6/93 to 6/96 = 3 yr. 
From June 24, ’96, to July 24, 96 = 1 mo., or 6/24 to 7/24 = 1 mo. 
From July 24, ’96, to Aug. 13, 96 = 20 d., or 7/24 to 8/13 = 20 d. 
2. To find the time from Sept. 14, 1891, to Mar. 11, 1895. Ex- 
plain each process. 
Process A. Process B. 
From Sept., 91, to Sept., "94 = 3yr., or 9/91 to9/94= 3yr. 
From Sept. 14 to Feb. 14= 5mo.,or 9/14 to2/14= 5 mo. 
From Feb. 14 toMar. 11=25d., or 2/14 to 3/11 = 25 d. 
3. What advantage has process B over process A? 4. Why is 
it well to know the months by their numbers as well as by name ? 
5. In process B, what is found at the left of the inclined line? At 
the right of it? 
6. Napoleon was born Aug. 15, 1769, and 


Process. died May 5, 1821. How long hact he lived? 
8/1769 to 8/1820 = 51 yr. Explain the process. 
8/15 to4/15 = 8mo. ; : ; ; 
4/15 to5/5 =204. 7. Find the time from May 22, 1890, to 


June 12, 1895. 
8. Find the time from Dee. 25, 18938, to Mar. 3, 1896. 
9. How long from 4/18, ’96 to 3/11, ’99 ? 
10. Find your exact age to-day. 


166 INTEREST. 


283.— Interest: Choice Meruops or ComputTinG INTEREST. 
of Methods. I. A general method, page 106. 


If. The bankers’ method, page 160. 
III. The one-dollar method, page 163. 

Any of these three methods may be used exclusively, but as no 
one method is always the best, 1t is well to learn to choose the one 
that will give an accurate result most quickly. 


Written. 


Solve the following problems by each method, compare results, and 
tell which method you prefer, and why. 


1. Find the interest of $ 360 at 7% for 207 d. 
2. What is the amount of $75 at 8% for 3 yr. 4 mo.? 
3. What shall be paid for the use of $723.60 for 85 days at 10% 


interest ? 


What is the interest under the following conditions ? 


Principal. Time. Rate. Principal. Time. Rate. 


4. $648. llld 4% 9. $432. leyr-3 38 anos es a 
5. $324. 167d. 5% 10. $767.80 3yr.11mo.9d. 534% 


6. $750. 200d. 9% ll. $50.40 10 mo. 41% 
7. $427. 93d. 12% 12° 3.8737) 114 d. 6% 


8. $865. 48d. 4% 13. $137.77 Ayr.9mo.25d. 74% 
14. Interest is the product of what three factors ? 


15. Which method of finding interest is best when principal, rate, 
or time is divided by 4? By 44? 6? Q9ori2? Why? 


16. Which method uses the aliquot parts of the time ? 
17. Which are “6% methods”? Why so called ? 


18. Which is the best method when there are years, months, and 
days in the time, and when cancellation is impossible ? 


19. What is the interest of $ 400 at 10% for 21 years ? 


EXERCISES. 


167 


284. — For FREQUENT PRACTICE. 


At sight. 

1. Find 200 months’ interest 
of $ 87.56 at 6%. 

2. Find 20 months’ interest 
of $ 300 at 3%. 

3. Interest of $500 for 4 yr. 
at 10% ? 

4. Interest of $1 for 6d. at 
6% ? 

5. Interest of $569 for 60 
days at 6% ? 

6. Interest of $100 for 4 yr. 
at 8% ? 


1. 2 yr. interest of $400 at 
9% ? 
8. 8 mo. interest of $200 at 
9% ? 
9. 22 mo. interest of $500 at 
6% ? 
10. 24 d. interest of $800 at 
6% ? 
11. 0.003 of the principal is 


the interest for « days at 6%. 

12. A principal gains as much 
as itself at 6% in x months. 

18. 11 mo. interest of $ 400 at 
9% ? 

14. A principal gains 25% of 
itself in # months at 6%. 

15. ;1, of principal =@ mo. 
interest. 


For dictation. 
1. 4 yr. interest of $500 at 
5% ? 
2. 34 yr. interest of $100 at 
8% ? 


3. 6 mo. interest of $120 at 


| 9% ? 


4. 7% interest of $500 for 


2 yr.? 

5. 60 d. interest of $ 567 at 
6% ? 

6. 20 mo. interest of $1200 
at 6% ? 


7. 10% interest of $1 for 
06 d. ? 
8. 6% interest of $1 for 17 
mo. ? 
9. 6% interest of $1 for 
112 days? 
10. 5 mo. interest of $ 240 at 
10% ? 
1l. At 6%, what part of the 
principal = 50 mo. interest ? 
12. 11 days’ interest is what 
part of a year’s interest ? 
138. 28% of the principal is 
how many years’ interest at 5% ? 
14. Find the interest of $600 
at 8% for 4 mo. For 
15mo. For 15d. 
15. $372 is the interest of 
$ 372 for how long at 6% ? 


For 5 mo. 


168 INTEREST. _ 


285. — Interest: Nore. — The method to be employed in the 
Choice of Methods. solution of the following problems is shown 
Written. by the Roman numerals J, JJ, or IIT (p. 166). 


1. What is the interest of $840 for9 mo. 17d. at 4%? 
2. Find the amount of $722 for 156 d. at 12%. J. 


3. What will settle an account of $425 that has been drawing 
interest at 5% for 5 yr.5mo.? III. 

4. May 17, 1893, I borrowed $ 284 at 21%. Aug. 15, 1895, how 
much interest had accrued? J/I. 


5. In 43 years how much will be received on a $5000 railroad 
bond paying 2% semi-annually? J. 


6. May 27, 1898, I paid a note of $ 475.28 that had been drawing 
4% interest since Dec. 31, 1894. JIT. 


Find by inspection the best method of solving the following problems, 
and use it in finding the interest. Try to forecast the result. 
7. $9000 on interest 7 mo. 24 d. at 4%. 
8. $728 draws 5% anereat for 20 months. 
9. 34% interest of $900 from Jan. 15 to Nov. 2. 
10. Principal, $72.59; time, 125 days; rate, 121%. 
11. What shall I pay for the use of $500 ror 50 days at 5% ? 
12. $320; 74%; July 7, 1845, to August 4, 1859. 
13. $720; 8%; October 19, 1890, to May 11, 1893. 


14 470 3305 © eligeds 20. 3% $872 4yr.8d. 
15. $648 41% 8mo. 21. 5% $ 5000 16 mo. 
16. $800 21% 1804. 99. 219, $178.91 104d. 
17. $950 9% 2484. 93. 8% $64.87 2954. 
18. $2000 7% 191 mo, 24.1% $3294 17% mo. 


19. $4000 4% 42 yr. 95. 419, $700 412 yr. 


RECKONED EXACTLY. 169 


286. — Exact 1. In computing interest for parts of a 
Interest : year we commonly consider 30 days a month 

+ « 26 OUVQa « T » i 1 
365 Days in an Inter- and 360 days a year. In taking 31) of a 
est Year. year’s interest to find the interest for 1 day, 


do we take too much or too httle, considering the actual length of a 
year ? 


2. Exact or accurate_interest is reckoned for the actual number of 
days in the given time, and counting 365 days to the year. It is 
used by the United States government and sometimes in other busi- 
ness transactions. It differs from common interest only as applied 
to parts of a year. What part of a year is August? February ? 
The last three months of 1896 ? 


3. ai, is what part of =4,? Explain this process : — 


eg ihe ee OO) 80 0 2272 
360 — 865 I 36 : 


4. If 1 day’s accurate interest is 72 of 1 day’s common interest, 
what is the accurate interest when the common interest is $ 146 ? 


ed el 
365 


5. If from the common interest I deduct +>, of itself I shall 
have the exact interest. Explain. 


6. Find the accurate interest of $ 500 
for 90 days at 4%. 


7. Find the exact interest of $1000 at 
5% from May 9 to Sept. 4. 


Common interest de- 
as ; 1 . . 
creased by =; of it- 


self is exact interest. 


Nore. — The exact number of days must be found; that 
is, 22+30+31+31+4=118. 


Find the exact or accurate interest of — 


8. $800; 6%; Aug. 11 to Oct. 9. 
9. $720; 8%; Jan. 4 to Mar. 15. 
10. $1200; 3 mo. 12.d.; 5%. 
11. $1500; 72d; 10%. 
12. What is the exact interest of $ 1000 for 2 yr. 8 mo. 9d. at 6% ? 
(Find common interest for 2 yr. + exact interest for 8 mo. 9 d.) 
18. Find the accurate interest of $5000 for 5 yr. 9 mo. at 8%. 


170 INTEREST. 


287. — Wholesale 1. Show the difference between grower 
or Retail. For Cash or producer, importer, wholesaler, retailer. 
or on Credit. 2. From which class of dealers do you buy ? 
Oral. 3. With whom do wholesalers have to deal ? 


4. The regular price of a pear-tree is $ 1.50. 
If I get it for $ 1.25, what is the discount or deduction ? 


5. If I buy a dozen at one time, I pay only $12. What per cent 
of the highest price is this? What is the rate of discount ? 


6. A man is trusted for goods billed at $100. He is to pay in 3 
months. How long is the term of credit ? 


7. The dealer offers to sell the same goods for ¥ 98 cash. Why is 
this? What per cent does he discount ? 


288.— Trade Discount. 1. My bill is $15, less 10%, as I am 
“in the same trade.” What must I pay ? 


2. Price per dozen, $2; for 30 dozen I pay $50. Without dis- 
count the cost would be $a. The discount was y%. 3. If I had 
bought 100 dozen, the net price, or what I actually pay, would have 
been only $1.60. Is this a larger or a smaller rate of discount ? 
Why ? 


4. Discount on a carload of coal is 10%, or $4. What would it 
be on 20 carloads at double the 
rate of discount ? List Price or). 


= Base. 


5. It is common tohaveaper- | Amount of Bill) 
manent list of prices and to change Discount = Percentage. 
the rate of discount as may be Net Price = Difference. 
necessary. 


List price, $40; net price, $32; rate of discount, x%. 

6. When the discount changes to 25%, the difference in price is y. 
7. $5 is 20% of list price. The net price is 2. 

8. Discount, 10% ; net price, $90; list price ? 


CASH DISCOUNT. ATE 


289.— Time and Oral.—1. A $4000 house is offered at $3500 
Cash Discounts. cash. The discount = $a. The rate=y%. 
2. Bought $200 worth of flour. If I need 
not pay for 6 months, what do I save? Explain. 
3. Which customer receives the larger discount, one who pays in 
3 months or one who pays ina year ? 
4. January 1 I buy $400 worth Discount is always a part 
of wool, and am promised a time or per cent of the price which 
discount of 2% if I pay by April 1. it reduces. 
By paying when the goods were 
bought, I should be allowed 4%. What is a cash discount ? 


Written. — Find the missing terms : — 


Amt. of bill. % off for Net cost. Discount for List price. %. 
cash. cash in 30 days. 

5. $2000 5 x 8. $24 $ 600 ny 

6. $900 x 810 oe es $ 150 1 

if x 2 490 10. $20 x 4 


290. — Successive 1. From 100 take 60%, from the remainder 


Discounts. 25%, from that remainder 10%, leaving x. 
Written. What per cent of 100 have you deducted ? 


2. A box of pens is listed at $1, but a retailer buys it for 50¢, 
the trade discount being 7%. 38. When he buys 100 boxes, he gets 
a second discount of 20% from the 
lower price, each box costing $y. Successive discounts are 
4. A third cash discount of 1% taken from the price as 
makes a box cost $z. 5. Have we already reduced. 
deducted 
(50% + 20% +1%) of $1 or 50% of $1 + 20% of 50 +1% of 40¢ ? 


Find net prices : — 


List. % off. List. % off. 
6. $15.40 20 then 5 9. $14.85 60, 10, and 2 
7. $49.50 50 then 2 10. $320.15 20, 5, and 1 


8. $600 45 then 3 ll. $4000 30, 12, and 3 


172 DISCOUNT. 


291. — Problems. 1. A library buys its books 35-% off. An 

invoice of $10,000 calls for how much net ? 

2. A bill is made—“ Terms: cash in 60 days.” What discount 
may be expected for cash at time of sale? (Money at 6 %.) 

3. When money is worth 12%, a dealer gives 4 mo. credit. 
What discount for cash may be expected ? 

4. Which would be more profitable in the end —to-sell for $ 100 
cash, or to charge $ 103, giving 6 mo. credit? Is the cash discount 
here more or less than 3 % ? 

5. Cash or net price, $760; discount, 40 % of .; list price, a. 

6. An invoice of jams is charged at $2500 on 6 mo, time, or 
with time discounts of 1-% a month. What amount will pay the 
bill in 30 days ? 

7. One buyer gets 30% off, another gets 25% andi%. Give 
net cost to each on a shipment of $ 2000 gross value. 

8. A merchant who gives 90 days on all bills allows 5% for 
cash. You infer that money is worth to him w % a year. 

9, A furniture maker allows 15 % from the list price. Find the 
net cost on an order amounting to $12,458, including $118 for 
carting, which is without discount. 

10. Tubing listed at $10,000 is billed less 60 % and 2 % for cash. 
Net price = a, 

11. The trade discount on certain goods is 70%. Large buyers 
receive a second discount of 10%, making the total discount x%. 

12. List price, $500; net, $425; discount, #; rate, y %. 

13. List price, $488.90; net, $591.12; discount, w; rate, y. 

14. 1800 ft. of moulding at 20¢, less 12 % to the trade and 1 %p 
for cash, cost... 

15. A shipment of sugar invoiced at $11,000 is subject to a rebate 
or reduction of 5%. Terms: 15 days. 414% off for cash makes 
the net cost 2. 

16. The discounts on a $ 1000 invoice are not 45 %, but are 30 %, 
10%, and 5%. Find the net price. 


INSURANCE. 173 


292. — Insurance. 1. If the owners of a hundred ships agree 
ear to share the loss 1f one ts wrecked, who might 


profit by the arrangement ? 

2. A man owns a house worth $3000. By spending $30 he 
can be sure that loss by fire will be made good. Many others 
do the same, and from their money his loss is paid. What % will 
he save ? 

3. An insurance company promises security in case of loss to 
those who have paid a certain per cent or premium on the insurance 
of their property. What will it cost to insure goods for $250 at 

1g? 
11%: 

4. 100,000 persons pay 25 ¢ each to an accident insurance com- 
pany. If it pays $15,000 in claims for injuries, and $4000 for 
expenses, the profit is a Who is the insurer 2 Who are insured ? 

5. The agreement to make good a loss on certain conditions is 
printed in a policy made by the wnderwriter or insurer, and held 
by the insured. The cost of a $ 25,000 policy at 3% a year is $ a. 

6. By paying an annual premium a person may be assured that 
at his death or at a certain age, his family, or he himself, will 
receive a specified sum. How will this money have been obtained ? 

7. A ship costing $210,000 is insured for 2 of its value at 2 %. 
If lost, the owners receive « The underwriters lose y. 

8. $40 pays for five years’ insurance on a brick store which cost 
$5000. The insurance valuation is $4000. What is the annual 
rate ? 

9. A wooden tenement house two miles from a fire-engine is 
insured for 3%, but only for one year. If the valuation is $4000, 
the cost for five years is w The property insured is called a risk. 
Compare the last two risks. 

10. Insurance provides for sharing loss due to what causes ? 

11. A schoolhouse is insured for 5 years at }% premium, which 
is $300. The insurance valuation is 2 of the cost of the house. 
What is the underwriters’ loss, if it burns ? 


174 INSURANCE. 


293. — Examples. 1. A stock of goods is worth $12,000. The 

Written. premium for a year is 1%, or $100. What 

is the insurance valuation? If destroyed, 

what will the underwriters pay, and what will the owner lose besides 
the premium ? 


Supply values for «:— 


Valuation. Rate %. Premium. Rate %. Premium. Insurance. 
2. $20,000 x $ 100 40) $ 12.50 z 
3. 3,900 x 42 6. & 62.50 $ 5000 
4. a $ 30 1. 2 AY 4TAO 


8. Why is property usually insured for less than its full value ? 
A $7500 house is insured at 11% for $62.50. The insurance valu- 
ation is @. 

9. A man is insured in a mutual company, sharing all gains and 
losses; $60 insured his $8000 house in full for five years. What 
was the rate? After five years $20 with interest was returned to 
him. This reduced the rate to a. 

10. A ship worth $30,000 is insured for 2 of its value at 21%. 
The possible loss to the owner, including premium, would be a. | 

11. $3.75 was the premium on # the value of some furniture at 
1% ayear. What was its insurance valuation ? 

12. One company offers to take a $12,000 risk at 14% for five 
years; another at 1% a year. Which is cheaper, and why ? 

13. $234.69 is the amount of a policy on some window glass. 
The premium is 2%; the difference, or $a, equals the value of the 
glass, which is ¥% of the amount of the policy. 

14. $2000 is 98% of the amount insured. Premium = 2% of a. 

15. If a stock of goods is worth $6930, what insurance at 1% 
will include that amount and the premium? 6930 = 99% of a. 

16. A block is insured for $5000 in each of 15 companies at 
an average rate of 4% for five years. Find the annual cost of 
insurance, not counting interest. What would each company pay if 
the building suffered $50,000 damage ? Why is it safer to divide a 
risk among several companies ? 


COMMISSION, 17d 


294.— Commission.  Oral.—1. I send goods to a person in town 
Selling through to be sold. I am the principal; he is my 
Agents. agent. He sells them for $100, and I pay 


him 13% for the service. He keeps $a as 
his commission, and sends me the net proceeds, which are $100 — $a, 
or $y. 

2. What is 2% commission on a sale of 3000 melons at 50¢ 
each ? 

3. My agent, a commission merchant in New York, sends me 
95% of what he collects, and keeps $20. He collects $ a. 

4. The gross proceeds, or sum col- 
lected, is $150. The net proceeds are Commission for sell- 
$ 147, less $15 expense of transporta- ing 7s a percentage of 
tion. The commission is $a, or y% | the sum collected. 
ae 


5. A consignment of goods to be sold is sent by the consignor to 
the consignee. The sum collected minus the 5% commission is 
$285. What are the net proceeds? What per cent does the con- 


a0 = 


signee return? 1% = , or $3. The gross proceeds are $ a. 


Written. —6. Four near! of peaches are consigned to a factor, 
or agent, who is to receive 5¢ a basket. The price realized is $1, 
but the charges for freight equal three times the commission. The 
owner receives what per cent of the gross receipts ? 

‘ 7. Net proceeds of a sale of cocoanuts, $100; charge for storage, 
$5.60; commission, $4.40; gross receipts, By; rate, «%. 

8. Find the commission on a consignment of rubber shoes sold 
~at 10% off $300, the consignee retaining 3%. 

9. A shipment of strawberries sells for $138.15, from which the 
agent pays out $125 for freight. His 5% commission amounts 
to $a. 


. . 4 ie Le ‘ 
10. Commission = 100 of gross, or of net receipts ? 


Commission + expenses + net proceeds = what ? 


176 COMMISSION. 


295. — Commission. 1. If I employ a correspondent, or agent, 
Buying through to buy goods for me, I must pay him a per- 
Agents. centage of the amount which he expends 


forme. If I send him $1000 with which 
to buy corn, $ 1000 1s my remittance to him. 
If he invests it all, how much more must I remit to him to pay his 
3% commission ? 


Oral. 


2. If I had sent his commission when I remitted the amount he 
was to expend, how much must I have sent for each dollar he was 
to expend ? 


3. If I send 100% of the amount he is to expend, do I send him 
enough to pay his commission? If I send him $1.05, on what sum 
will he receive 3% commission? Why should the remittance be 
105% of the amount to be expended ? 


4. Should an agent receive a commission on his own pay, or on 
only so much as he expends for his 
employer ? Commission for buying is 

6. An agent buys $ 2000 worth of | @ percentage of the amount 
copper, charging 1% commission, or | ¢*Pended. 
bw. What must his employers re- 
mit to pay both bills? For every dollar he spends they must remit 
exactly _.. 


6. Remitted $102.50 to buy umbrellas; commission 24%. How 
much of my remittance will buy $1 worth of umbrellas and also 
pay the agent? The agent can buy $a worth. 


7. An agent charges 12% for buying chair stock. If he is to 
spend $ 1000, how much must I remit? 


8. Remittance, $400; goods purchased by agent, $337.50. He 
pays $25 for storing and forwarding them. The gross amount ex- 
pended is On what sum should he receive his 10% commission ? 

9. Remittance $9.45, less 5% commission = amount of purchase. 

Explain: $9.45 + $1.05 = a, - 


FOR BUYING. LTT 


296.— Commission for 1. At 5% commission, how many 
Buying. dollars’ worth can be bought for $ 126, 


leaving enough to pay for the service ? 

2; The principal in a_ transaction 
remits $2050 for a purchase of apples, less 2}% commission. 
$ 2050 is a % of the amount to be invested. 1% of it = y, 100% 
Oru. %, OF ax 

3. The agent above mentioned spends only $1500. How much 
of the remittance must he retain ? 

4. 11% or $ 20 is a charge for buying hides. Find the base and 
the entire remittance. 


5. When the commission is 13%, the amount expended is what 
100 x 4 __ 


1018 x4 


Written exercise. 


fraction of the remittance due ? 


_— 


In Selling : 
Amount collected or | 
Gross receipts, bah 
Commission = Percentage. 


In Buying: 


Gross amount | 
expended  } 


Commission Percentage. 
Base 
Remittance + 
Commission. 


Net Commission 


ob 
l Expenses. 


r 
\ 
| 
) 


Proceeds = Base — - 


297. — Examples. 1. $927 is the amount sent to purchase 
granite and pay the agent 3%. By what 
Written. must you divide $927 to find 1% of the 


base? What is the agent’s commission ? 

2. Remitted $ for the purchase of glassware. The commission 
was $ 42, or 31% of $y. 

3. Sent my agent $1200 for purchasing wheat. His commission 
is 5% of the purchase, or $ a. 

4. Forwarded $ 287.50, and received in return goods worth $ 250. 
The commission at 1% would have been $2. Actual rate, 7%. 

5. A person bought from my agent $547 worth of straw. When 
a 2% commission and $47 expenses have been deducted, I receive $a. 


178 COMMISSION, 


Supply values for x and y (dealings with selling agents): — 


Gross Proceeds. | Expenses. — Rate of Commission. 
6. $437 0 2% 
7. a $ 47.50 1% 
8. $ 250 x y 
9. x 0) 2% 
10. $1200 $ 100 a 
1l. $1680 x 4% 
12. a 0 y 


298. — Examples. 


Written. 


Commission. 


Net Proceeds. 


wv Yy 
$ 27.50 y 
$ 25 $ 200 
y $ 980 
$ 18 y 
y $ 60 
$ 13.52 $ 437.18 


1. $4380 is received from a sale of linen. 
After retaining 14 % commission and paying 
$ 2.50 for advertising the sale, what is the 


balance to be remitted ? 
2. What value of goods can be bought on 5% commission from 


a remittance of $577.50, allowing $ 24.50 for advance charges of 


forwarding the purchase ? 


3. A correspondent retains 44% on the receipts of a certain 
sale, and after paying $4.37 for carting, etc., remits $200. The 


gross receipts include $2 + y+ $z. 


4. A dealer sends to his agent $ 20,500, including a commission 


of 2% on what the agent will spend, which is $100 for insurance 


and sundries, and the balance $ a for wool. 


Supply values for « and y (dealings with purchasing agents) : — 


Remittance. cebe gamed 
5. $595.80 yy 
6. $179.76 $148.30 + $ 27.94 
if x $ 755 
8 a y 
9. $1293.75 . 
10. % $ 684.10 + $ 88 


Am/’t of Purchase + Freight, R 


ate of Commission. | Commission. 
y $ 6.60 
v y 
y $ 22.65 
5 % $ 9.90 


PROBLEMS. 179 


299. — Problems: 1. On a bill for hardware amounting to 

Discount, Insurance, $480, I received four successive discounts 

Commission. of 10% each. What is the amount to be 
paid ? 


2 


2. My residence is insured for 2 its value in the Provident 
Insurance Co. at }%. The premium is $40. What is the value of 
my property ? 

3. My agent in Mobile bought 40,000 lb. of cotton at 97%. His 
commission is }% and his expenses are $143.75. What shall I 
remit him ? 

4. A real estate broker sells a farm for $8000 at a 5% com- 
mission. What are the net proceeds of the sale, and what is his 
commission ? 


5. I can buy 1000 bbl. of oil at $1.12} with 3 % off in 30 days, 
5 % for cash. What shall I save by oe the better offer, 
money being worth 6 % ? 


6. The estimated loss of property at a large fire was $ 275,000. 
The insurance received was $180,000. How much must be taken 
in new risks at an average of 2 % to cover this loss to the under- 
writers together with $ 5000 expenses ? 


7. I receive from my agent in London a draft for $3860, the 
net proceeds of a sale of flour at 351% commission. What were 
the gross proceeds ? 

8. A drummer earns $2500 annually. $1000 is a guaranteed 


salary; the remainder is his commission of 5%. What are his 
annual sales ? 


9. A broker negotiates a loan of $6500 on a real estate mort- 
gage. His commission of 2 % and the expenses of examining title, 
etc., are $72.37. What does the mortgager receive ? 


10. Bought 1000 gross of screws at 27 cents, with a discount 
of 15,10, and 5. I sold the lot at cost plus 30%. What was my 
gain ? 


180 INTEREST. 


300. — Promissory May 17, 1895, Edward Rich lends to Thomas 

Notes. Poor $180 to be repaid when the lender asks — 

it, together with interest at 5%; as evidence 

of the loan and security for its payment the lender receives from the 
borrower a promissory note like the following: 


/Haneneter, Nay ff ad Oe 
Qn demand, after date, %. promise to pay to the 
_Gdward Reh 


One /fundred Gighty 


order of 


with interest, at five per cent. 


Value received. A 
Shomae Loor. 


1. Who is the maker of this note; i.e. the promissor ? 
2. Who is the payee, or the one to whom promise of payment is 


made ? 


3. What is the face of the note, or the sum namedin it? 4. When 
is the note payable ? 


5. In what way does the maker acknowledge the receipt of the 
note, or its equivalent ? 

6. Why is such a note as this called a demand note ? 

7. Why is it called an interest-bearing note? 


Promissory notes are a kind of property, and may be bought and 
sold like other property. 

8. Show that on May 17, 1896, the above note is worth $189 to 
the owner or holder. What will the owner gain or lose by selling it 
for $ 200, 2 years from date ? 


PROMISSORY NOTES. 181 


9. Whenever the payee of a note 
transfers it to the ownership of another 
person he first indorses it; that is, he 
places his signature on the back of it. 
6dward Reh. What is an indorser 2? An indorsement? 

10. The payee of a note may indorse 
it in blank as in A, or he may make a 
special indorsement, as in B. 

A. blank indorsement makes the note 
payable to the holder. A special indorse- 
ment makes it payable to the person 
named by the indorser as payee. Copy 
and indorse the note of Thomas Poor. 
Say to the order of Every indorser of a note is responsible 
for its payment unless the words “ with- 
out recourse” precede his signature. 
6dward Reh. 11. The holder of the note on the 
preceding page demands payment of 
Mr. Poor Aug. 27, 1897. What is the 
amount then due ? 


/remry K¥all 


12. If the words “one year,” “four months,” “sixty days,” ete., 
were substituted for “on demand,” when would the note be payable ? 

Nore. — In some states three extra aye after the expiration of the time named in the note are 
allowed the maker for its payment. They are called days of grace. Interest is exacted, how- 
ever, for the days of grace, [See § 310.] 

13. If the note were a four months’ note, at what date would it 
be payable without grace? With grace? When, if it were a 2 
months’ note? A 6 months’ ? 

Notes mature, or are legally payable, on the day when the time 
named in them expires, or on the third day thereafter, when grace is 
allowed. 

All notes that contain the words “with interest” draw interest 
from date unless otherwise specified. All other notes draw interest 
from maturity. When no rate of interest is specified, the legal rate 
is understood. [See Appendix, p. 15. ] 


182 INTEREST. 


301. — Promissory Notes. Make interest-bearing notes answer- — 
ing to the following conditions, and 
compute the amount due at settlement. 
In finding the day of maturity, allow three days of grace. | 


Written Exercises. 


Date. Face. Time to run, Payee. Rate. Settled. 


3/17,’95 $240 Ondemand A. P. Rice 6 9/14, ’96 
8/12,’96 $800 One year E. F. Foss C1275 oo 
4/21,’96 $725 Fourmonths Wm. Ward 5 Maturity 
6/15, ’93 $1800 Six months Thos. True 4 9/21, 796 
1/19,’95 $610 ‘Two years A.M. Bates’ 9.5 )12/25on 
2/24, ’96 $280 Ondemand Rk. E. Nye 4i 7 /16,’98 
eo $75 Sixty days E. B. Hale 12 Maturity 


Se oe ace ome aaa 


8. Write a note without signature, making yourself payee, and 
transfer the note, properly indorsed, to your teacher. 
9. A note matures Aug. 16, 1897. Its face is $1200, and it has 
been running at 4% since May 12,1894. What is paid at settlement ? 
10. Draw up a note in which Thos. Talbot hires $1700 of Samuel 
Strong, at 4%, agreeing to make payment on demand. Demand for 
payment is made May 14, 1895, the note being made March 12, 1893. 
What amount pays the debt ? 


302. — Partial Payments 1. I pay $80 on a note whose face is 
of Notes. $150. What part do I pay ? 
2. I pay 30% of a note of $1500. 
What is the partial payment ? 

8. A note of $300 draws 10% interest. What amount would 
discharge the note at the end of the first year? Suppose that 
instead of the note being paid in full at that time a partial payment 
of $100 were made, what would then be due ? 

4. Would the $100 pay all the interest due? How much of the 
face or principal would it also pay ? 


PARTIAL PAYMENTS. 183 


5. How much of the original $500 does the maker of the note 
continue to keep? On what sum, therefore, should he pay interest ? 

6. Ifthe remaining $ 230 should be used another year, the interest 
on it at10% would be $a, and the amount due would be $ 230 + $ a, 
or Sy. 

7. If another partial payment of $100 should then be made, a 
remainder of $ ¥ — $ 100, or $z, would still be left at interest in the 
hands of the maker of the note. 

8. Give the values of a, y, and z in the solution of the following 
problem : 

On my note, payable to you for $300, I make a partial payment 
of $100 at the end of each year for three years. What is then 
due you, 10% interest heing charged ? 


SOLuTION. 

I. Of yourmoney Ihave foruse ..... . $800 
For a year’s use of itat 10% ITowe . .... . x 
At the end ofthe yearIoweyou. .... . . $380 
I make a partial payment to you of ee. __ 100 

II. This leaves a balance for me to use Dt ri ee SoU 
For a year’s use of this sum I owe you . __ 23 
At the end of the second yearlowe you .... $ y 
I make a second partial payment of . . ... . 100 

III. Inow have of yourmoney only ..... $ @ 
For a year’s use of thissum ITowe you... . . 15.30 
I owe you at the end of the third year . . . . . $168.30 
PE Paes OU se Teen wg ete be per cy.” 100 

PVeeUalialh Btu OWe VOU...) es ne eg oh Ba OB.80 


9. I borrow $500, giving my note at 6%. At the end of each 
year, for two years, I pay $100. How much remains due? 

10. A note for $800 draws 5% interest and is dated May 1, 1894. 
May 1, 1895, $200 is paid; May 1, 1896, $100 is paid. What is 
due May 1, 1897 ? 

11. Two partial payments of $ 400 each are made on a $ 1000-note, 
dated Aug. 20, 1895. The rate of interest is 5%. The first pay- 
ment is made Aug. 20, 1897, and the second, Aug. 20, 1899. What 
amount remains due ? 


184 INTEREST. 


When partial payments of a note are made, 
the owner records the amount and date of 
each on the back of the note. 

To find the amount due on the following 
note Dee. 31, 1901: 


303. — Partial Pay- 
ments of Promissory 
Notes. To find 

Amount Due. 


§ 720 — Shringpretd, Aug. /4¢, 1895. 

On demand after date 4% promise to pay to 
te Ake LE al terry SYoward Y Eo. 
seven /fundied Swenty Dollars 


the order of 
with interest at a2 per cent. 


Value recetved. ¥. Bae 
Jhomar SF. Sowell, 


INDORSEMENTS ON Back OF NOTE. 


The Supreme Court of the 
United States has decreed 
that, — 


Reeetwed on the within note: 
Dee. 26, 1896, § 209 
Sept. /¢,/899, 175 


Partial payments of notes 
must first be used to cancel 


dee. 3/, 1900, 4.00 the interest due. Any bal- 


Settled, Lee. 3/,/90/. 


ance remaining may be used 
to lessen the principal. 


1. Who puts these indorsements on the note? 2. Will any other 
receipt be requested for the $200 paid Dec. 26, 1896, and if so, 
by whom? 8. What needs to be done before the note can be trans- 
ferred to a third person ? 


PARTIAL PAYMENTS. 185 


SOLUTION. 


From date of note to Ist payment. 


8/14, ’°95to 8/14,°96 =lyr. . . .* $0.06 $ 720 
5/14, 96 to 12/14, "06 =-4'mo. . .)\. «. 0.02 0.082 
12/14, 96 to 12/26,°96 = 12d. . . . . Daa 44 
Intol Serie .eee S52 8 0;088 57 6 
Interest due when lst paymentismade ...... . $59.04 
Beeee: Rote, Or lst principal <u. 2%) aoe aE See sy tes 720. 
Amount due at time of lst payment .. . Pees Pe $779.04 
Ist payment, which cancels interest due, and more .. . 200. 
Remainder which continues to draw Int.; 2d Prin. $579.04 
From Ist payment to 2d payment. 0.1652 
12/26, °96 to 12/26,°98 = 2yr. . . . . $0.12 9650 
$B f20, eto 07 20,00 == 5000.1. oy 0.04 1 73712 
8/26, 99: to 9/14,°99=19d. . . . . _ 0.003} 34 7424 
$ 0.1634 57 904 
Interest due at time of 2d payment . . ....... $ 94.48002 
2d principal. ... ed oe" err ee 579.04 
Amount due at time of 2d PAYINCNGr sh." 4S oi eee $ 673.52 
2d payment cancels interest and part of Oenehtal av 1 tab 175. 
Remainder which still draws Interest; 3d Prin. $ 498.52 
From 2d payment to 3d payment. 0.0772 
9/14, 1899 to 9/14,1900=1yr. . . . $0.06 6)249260 
9/14, 1900 to 12/14,1900=3mo. . . . 0.015 41543 
12/14, 1900 to 12/31,1900=17d. . . . 0.0028 2 48964 
30.0773 34 8964 
Interest due at time of 3d payment. . . ..... . $ 38.80147 
3d principal. . . . ate PY res. keene 498.52 
Amount due at time of 3d Seon Oe calle $ 537.32 
3d payment cancels interest and part of aaerial 5y Sel Ae 400 
Remainder which still draws Interest; 4th Prin. $ 137.32 
From 3d payment to settlement. une 0.06 
12/31, 1900 to 12/31, 1901 =1lyr. . . $0.06 Int. $ 8.2392 
4th principal ... . Tee ay ee ee ae 137.32 
Amount due at pertiament ee tity 8 et ie Bl $ 145.55 


4. How much interest was cancelled by the first payment? How 
much of the principal? 5. How much of the third payment was 
apphed to lessen the sum on which Powell was paying interest ? 
6. What interest is due Sept. 14,1899? At settlement ? 


186 PROMISSORY NOTES. 


304. — Partial Pay- Commonly the partial payment of a note 

ments too Small to will not only cancel all the interest due, but 

cancel Interest will also pay a portion of the principal, and 

Due. thus reduce the sum at interest. It some- 

times happens, however, that the payment 

is too small even to cancel the accrued interest. In such cases 

interest must not be used to increase the principal, which must 

never represent more than the money actually and previously due 
to the creditor, and in use by the debtor. 


To find what is due at settlement on a note of $600 dated Aug. 
15, 1895, drawing 6%, and indorsed as follows: 
Dec. 15, 1894, - $ 25. 
Sept. 15, 1896, $ 200. 
Settled Aug. 15, 1898. 


SOLUTION. 
From date to Ist payment. 
8/15, °93 to 8/15, °94=1 yr. $0.06 
8/15, °94 to 12/15, °94=4mo. 0.02 $600 
$0.08 0.08 


The payment of $25 will not pay the accrued interest, $48.00. Hence we 
compute the interest — 


From date to 2d payment. 


8/15, 393.1087 109051) yrs ee eee OLS 
8/40,200 10 07 1b. ¢ S05 Ln0. awe ee 0.005 $ 0.185 
Interest of $1 et sien 600. 
Interest due at 2dpayinents- a), 2) nn he ee $111.0 
Face of note on interest . .. . gee ae 600. 
Amount due at 2d payment. . . al $711 
Sum of Ist and 2d payments ($25 1 $ 200) . La 225 
New Principal on Interest. .. . 5 ee $ 486 
From 2d payment to veltlenient 0.115 
9:/15)796 to. 9/155 O7cd-yr. = ee 0.06 2 430 
9/15, ’97 to 8/15, US a 1) Oe ef eee 0058. 86 
0.115 486 
Interest due at settlement ...... ae $ 55.890 
Principal due at settlement. . . . ee 486. 


Amount due at eat ee he $ 541.89 


PARTIAL PAYMENTS. 187 


1. Why do we not add the first interest, $48, to the principal, sub- 
tract the payment, $25, and compute the interest on the remainder, 
$ 623, as in the example on page 185? 2. Will the two payments 
together cancel the interest due? How much of the principal be- 
sides? 8. How must we proceed when a payment will not cancel 
the interest due ? 


4. Show why the debtor might object to the arrangement pro- 
posed below : 


ieloaty MY Jamies wes. 1s fe . . 2. te 600 
A years interest'at5% is... . . . 30 
Hethenowesme ........ . $630 
He paysme... ? 3 eee er 20) 
I ask him to pay me Pr ReReey OTE Aa: 5 610 


5. What will settle a 5% note for $1000 Aug. 1, 1898, on which 
$ 50 was paid 2 years after date, and $500 4 years after date? The 
date was Aug. 1, 1890. 

6. What was due May 1, 1899, on a note for $5000, on which 
$ 100 was paid at the end of each year for 3 years? The rate was 
4%, and the note was given May 1, 1895. 

7. Cyrus Drew gave Frank Watson his note for $800 at 3% 
interest, Aug. 17, 1892. Dec. 23, 1894, he paid $300, and May 29, 
1896, he paid $40. What was due six years from date ? 


305. — Promissory 1. A note of $300 drawing 6% interest 

Notes: Partial and dated May 15, 1892, has a single indorse- 

Payments. ment of $200 paid Aug. 15, 1894. What 
amount will settle the note June 21, 1896 ? 

2. Oct. 19, 1894, Silas Deane pays $275 
on a $1000 note which he gave Charles Dole May 19, 1892. The 
note draws 4% interest and is settled Feb. 19, 1896. What is the 
amount due ? 

3. A demand note for $4000, drawing 3% interest, has a $ 1000- 
payment made upon it at the end of the first and of the second 
year. What will settle it at the end of the third year ? 


Written exercise. 


188 PERCENTAGE. 


4. The face of a 5% note is $1200. It is dated Jan. 1, 1894, 
It has the following indorsements : 

May 138, 1895, $ 400. 
Aug. 25, 1897, 300. 
What is due at settlement, Aug. 25, 1899 ? 

5. John Haley pays $400 July 20, 1896, on a note for $625, 
dated Oct. 20, 1895. Dec. 36, 1896, he pays $120. What is due 
on the note four years from date, the rate of interest being 9% ? 

6. Horace Swan buys a 160-acre farm in Idaho for $17 an acre. 
He gives a 4% note for 3 the cost and pays $680 cash. At the end 
of 2 years he pays one half the face of the note, and at the end of 
3 years he pays one half the remainder. How much is still due 
six years from the date of purchase ? 


306. — Customs or The expenses of the national government 
Duties. are paid from — 
I. The Internal Revenue, chiefly taxes of a 
fixed sum on the right to make or sell spirituous hquors, tobacco, ete. 
II. Customs or Duties, which are taxes on goods imported from 
foreign countries. 


1. What need of money has the government? Mention several 
articles commonly imported. 

The Free List includes all classes of goods that are exempt from 
duty. The Tariff is the list of dutiable goods with the rate assessed 
on each kind. 

Ad valorem duties are a percentage on the cost of the goods where 
they were bought, as shown by the Invoice. Specific duties are fixed 
according to number, quantity, weight, etc.,—not according to 
value. 

2. Find the duty at 15% on 200 T. of coke invoiced at $1.50 a 
ton. Is the duty specific or ad valorem ? 

Note. — Invoices are made out in the money of the country where the goods were bought. 


When changed to United States money, the duty is computed on the nearest dollar, 50 cents 
counting as $1, 


CUSTOMS AND DUTIES. 189 


Imported goods must be brought to a Port of Entry, a place where 
the government has established a Custom-house with officers for the 
collecting of duties. 

3. What is meant by smuggling ? 

4. A gross of leather pocket-books invoiced at $12.50 a dozen 
pays 80% ad valorem, or $ a. 

Tare, Leakage, and Breakage are allowances for boxes, bags, etc., 
used in packing, and for liquids lost from barrels or bottles, ete. 

5. A dealer withdraws from the custom-house 4 'T. (2240 lb.) of 
rice; 8% is tare. On what weight must he pay duty? What will 
it amount to at 1}¢a pound? 6. Is this a specific or an ad valorem 
duty, and why ? 

7. Define gross weight and net weight. 

8. An importation of velvet invoiced at 5181.35 francs weighs 
400 lb., 10% tare. Which would be more, a duty of $1.50 a pound, 
or 50% ad valorem ? 

9. 20 bbl. vinegar of 42 gal. each pay a specific duty of 7i fa 
gal., leakage 50 gal. Find the whole duty and the per cent of 
leakage. 

10. The duty on cut nails is 223% ad valorem. $183.75, the cost 
including duty on a certain importation, is what per cent of the 
amount of the invoice? They were invoiced at $ a. 

11. In 1888 the duty on brussels carpets was 30 ¢ a sq. yd., and 
30% ad valorem. Find the total cost of 200 running yards invoiced 
at 6s. a yard, $ yd. wide. 

12. The gross cost of a lot of calf-skins is $ 740, including $ 20 
freight and $120 duty. Find the ad valorem rate. 

13. Gross cost $1315, freight $50, duty $165, rate a. 

14. If the cost of gloves is doubled by importing, what will be the 
profit on a pair invoiced at 60 fr. a dozen, and sold for $ 2.25? 


307. — Taxes. The expenses of states, counties, cities, and 
towns are met by levying taxes annually on 
adult male citizens, and on all owners of property. 


e 


190 PERCENTAGE. 


Oral. —1. Why are some persons taxed more than others ? 

2. Mention several ways in which taxes collected in your city or 
town are expended. 8. What need have county and state to raise 
money by taxes? 4. Find the amount of poll taxes in your city 
or town. 

Property is divided into two classes for taxation: Real Estate, 
regarded as immovable, as land and buildings, including mines, 
quarries, forests, railroads, etc.; and Personal Estate, which is usually 
movable. 

5. Give examples of valuable personal property. 6. How large 
a tax on property is assessed in your city or town ? 

Suppose a town requires a certain sum of money for the expenses 
of a coming year. Officers, called Assessors, first estimate the value 
of the property to be taxed, and then assess each owner in proportion 
to what he has. 

7. If the amount to be raised is $20,000, and the poll tax 
amounts $1600, how large a property tax must be assessed ? 

Written. —8. Assessors find the value of real estate in their city 
to be $15,000,000, and of personal estate $5,000,000. .They assess 
a tax for state, county, and city. If the rate is $15 tax on $ 1000 
valuation, the total to be collected will be a. 

9. The rateis 124 0n 1000. What will be assessed on $ 4,000,000 
valuation? A man who pays $125 is assessed on how much ? 

10. A man’s tax, including $2 poll, is $122. The rate is 11% 
or x on $1000, y on $1. His property is valued at how much ? 

11. 900 is what per cent of 60,000? If $12 is the tax on $1000, 
what is the tax on $1? what the rate per cent ? 

12. Find the rate when the tax on $120,000 is $1770. On $100 
the tax would be a. 

13. If 154% =z per thousand, how much can be raised on a valu- 
ation of $ 15,492,500 ? 

14. Valuation, $40,000,000; property tax, $640,000; rate, o. 

15. Property taxes in a certain town are $25,100; an estate of 
$15,000 pays $195. Find the rate and the total valuation. 


COMPOUND INTEREST. 11 


16. A person is assessed in one town on $3500 real property, 
where the rate is 11% ; in another on $ 8500, rate $14.50 per $ 1000; 
in a third on $5000 personal estate, @ 14 on a dollar; besides a 
$ 2 poll tax. What does he pay in all? ; 

17. Valuation, $11,000,000; property tax required, $121,000. 
Each dollar must pay a tax of a If the cost of collecting the tax 
is 1%, and the poll tax $1100, what are the net proceeds ? 


308. — Compound 1. Jan. 1, 1895, Mr. D. borrows $500 of 
Interest. Mr. C., agreeing to pay 6% interest at the 
end of every year. How much interest is due 

at the end of the first year ? 

2. Who is entitled, under the agreement, to the use of this $30 
interest for the second year, the debtor or the creditor? 3. If the 
debtor uses the overdue interest during the second year, instead of 
paying it to Mr. C., is it just that he should pay for its use? 4. On 
how much, then, should the debtor pay interest for the second year, 
including both principal and overdue interest ? 

Interest reckoned on both the principal and the overdue interest 
added to the principal as often as due 1s Compound Interest. 


Notre. — When interest is added to the principal, or ‘‘ compounded,” it is done yearly, unless 
otherwise stated, as half-yearly, quarterly, or oftener. 


5. What is due on a debt of $600, which has been standing 2 yr. 
6 mo. 18 d., interest at 6% compounded annually? Explain the 
following process. 


Principal used for lst year . . . . . . §600- 
Interest due atend of Ist year. ... . 36. 
Principal used for 2d year . . . . . . $686. 
Interest due atend of 2d year. ... . 38.16 
Principal used for6 mo. 18d... . . . $674.16 
Interest due atendof6mo.18d.. . . . 22.25 
Amount due at settlement . . . . . $696.41 
BiPSG Tire) oe a rs, ee ls, os 600. 
Compound interest. . .... .. §$ 96.41 
Simple interest would have been . . . 91.80 


Interest on all overdue interest. . . . $ 4.61 


192 INTEREST. 


ond 


6. Find the compound interest of $500 for 3 yr., at 5%. 

7. What is the compound interest, at 4%, of $2000 for 2 yr. 
6 mo. ? 

8. Interest compounds semiannually on $400, at 8% a year, for 
1 yr.6 mo. What is the amount due? 

9. Interest compounds quarterly for 9 mo., at the annual rate of 
4%. What is due, the principal being $1000 ? 


Nore. —Compound interest is generally computed by means of tables. [See p. 14 of Appendix. ] 
The collection of compound interest on notes and debts cannot be enforced, even though agreed 
upon. 


10. Principal, $480; time, 1 yr. 3 mo. 15d.; annual rate, 6% 
payable half-yearly. 


~ 


309. — Savings- 1. What advantage does a thrifty person 
Banks Deposits. receive from depositing his savings in a bank ? 
2. What use do savings-banks make of the 
money received from deposits ? 8. How is their profit made ? 
4, In 1894 there were in this country 1006 savings-banks, with 
4,739,194 depositors, and deposits amounting to $ 1,739,006,705. 
What was the average to a depositor ? 


Nore. —Savings-banks allow compound interest on all deposits remaining for a full interest 
term. They generally compound the interest or declare dividends semiannually. 


Find what maybe withdrawn from the Franklin Savings-bank by 
depositors under the following conditions : — 
5. James Swan; $400; 4%; 1 yr. 6 mo.; semiannual dividend. 


6. Edw. Wise; 250; 4%; 2 yr.; semiannual dividend. 
7. Sam’] Rand; 800; 2%; 9 mo.; quarterly dividend. 
8. John True; 3825; 4%; 7 yr.; semiannual dividend. 


[Use table in Appendix for Exercise 8.] 


310. — Promissory 1, At the present time, August, 1895, in 
Wotes: the Day of all but ten states and territories the law 
Maturity. allows the maker of a note three days of grace 
for its payment in addition to the time named 

in the note. Why is this? [Appendix, p. 17.] 


Norr.— The ten states that allow no grace are Vermont, Connecticut, New York, New Jersey, 
Illinois, Wisconsin, Idaho, Oregon, Utah, and California, 


——s Oe Se 


BANK DISCOUNT. 193 


2. A note matures, or is legally due, on the day when the time 
named in it expires; or, on the third day thereafter when grace is 
allowed. What if the three days expire on a Sunday?  [ Appendix, 


p. 17.] 


3. When a note is payable “ months after date,” calendar 
months are understood, and the note matures (without grace) on the 
corresponding day of the month, or on the last day of the month 
when there is no corresponding day. Show that a 2-months note, 
dated either Dec. 28, 29, 30, or 31, will mature without grace Feb. 
28 in a common year, or Feb. 29 in a leap year; and that if grace is 
allowed, it matures March 3. 


4. When a note is payable “ days after date,” the exact 
number of days mentioned must be used in finding the day of 
maturity, plus three days if grace is allowed. Show that a 60-days 
note dated Feb. 15, 1895, matures Apr. 16/19. 


Find the date when notes mature, if — 


Payable Running Dated Payable Running Dated 

5. Mass. 2 mo. Dec. 31 ll. Vt. 6 mo. Aug. 31 

CaN Y. 60 d. July 10 12. Cal. 60 d. Feb. 17 

yew ir, 3 mo. Feb. 28 13. Ill. 90 d. Aug. 17 
8. Penn. 90d. Nov. 30 14 N.J. 30 d. June 7 

9. Wis. 4 mo. July 31 15. Minn. 4mo. Oct. 31 

Ba. bh 0) d. Jan. 20 16. Conn. 2 mo. Feb. 29 


311.— Bank Dis- 
count: an Allowance 


[Review pages 180, 181. ] 
1. What responsibility does the indorser of 
toa Bank for the Pre- 4 note assume ? 


ayment of a Note. ; : : 
od 2. What risk does a bank take in buying 


notes made by reliable persons and properly indorsed by the payees ? 
3. Show that banks cannot afford to pay their face value for such 
notes. 


194 PROMISSORY NOTES. 


& 500___ Roeheter, oS. yp, Aug. 1/2, 1895. 
Iwo montha after date, J promise to pay to the 
arder of. Antahee ee APY THORGAN Sg a ee 
PVE ST UNAVER 8 Vi ta tee ee Dollars 


at the YSeeond ational Bank. 


Value received. 
| Yamee Shaw. 


"hee Boe Dk ee a a ie a atte et a 
4. Who must indorse this note before it passes into the ownership 


of a third person? 6. Who is to pay the note? Where? 6, In 
what case may Mr. Morgan be called upon to pay it ? 


7. When does it mature? 8. Is grace allowed? 9. What is its 
value at maturity ? 10. Why is it not worth $500 at date ? 


ll. Aug. 12 the holder of this note offers it for sale to the Second 
National Bank. The bank officers are satisfied that either Mr. Shaw 
or Mr. Morgan will pay the note at maturity. The current rate 
of interest is 6%. According to the custom of banks they compute 
the interest at this rate on the face of the note for 2 months, the 
time for which the bank will lose the use of its money. They find 
this interest to be $a. They deduct it from the face of the note 
and pay the seller the remainder, $495. What occurs in connection 
with this note on October 12th ? 

The allowance or discount made to the bank by the holder of the 
note for its payment before maturity is the Bank Discount. 

The sum received for the note from the bank is the proceeds or 
avails of the note. 


12. When does the bank get back its $495? When does it receive 
the $5 paid for the use of its money? If it had lent $495 for 
2 months at 6% interest, how much would it have received? The 
charge of $5 was 1% of x, though the money it lost the use of 
was ¥. 


BANK DISCOUNT. 195 


Should Mr. Shaw fail to pay the note before the closing of the 
bank on the day of maturity, immediate notice is given to the indorser 
by a notary public, and he is held for payment. 


13. I sell a 4-mo. note for $600 payable without grace, to the 
Marine Bank. Money is worth 5%. What discount from the face 
do I make to the bank? What are the proceeds of the note ? 


14. A note for $1200 is bought by a bank. The money will be 
repaid in 90 days, with grace. The rate of discount is 4%. What 
does the bank pay for the note? How much interest will it receive 
for the use of its money? When will it receive this interest ? 


15. What is interest ? When I sell a note to a bank, is the allow- 
ance that I make reckoned on the money that I get to use, or on more 
or less? How does the bank discount differ from interest? In bank 
discount is the cost of the accommodation paid in advance, or is it 
paid when the money used is paid? Who pays the bank discount ? 


312.— Bank Discount: Most of the notes discounted at banks, or 
Notes discounted by brokers or others, are given for short 
at Date. times, 30, 60, 90 days, or 2, 3, 4, or 6 months. 


Nore.— Compute bank discount as if it were interest on the face of a note for the time the 
bank’s money isused; and take the shortest method. Allow three days of grace if such is the 
lawful custom in your state. Answers are given both with grace and without it. 


1. What will a bank pay me for a note of $800 payable in 3 
mo., the rate being 44% ? 

2. How much of its money will a bank permit me to use for 6 
mo. in return for a note of $720? Money is worth 5%, and the 
note runs 60 days. 


3. What allowance shall Mr. Strong make to the Exchange 
Bank for its prepayment of a 4-mo. note for $875 at a discount rate 
of 3% ? 

4. A $600 note dated July 5 is to be paid Sept. 5. Required 
the proceeds if discounted at 5%. 

The term of discount is the time for which the bank’s money is 
used. It extends from the day of discount to the day of maturity. 


196 PROMISSORY NOTES. 


~ 


Find the bank discount and the proceeds of notes made under these 
conditions, and discounted at date : — 


Face. Time Rate of Face. Time tate of 

to run. Discount. to run. Discount. 

5. $525 30 d. 3% ll. $842 4 mo. 2% 
Oe elt 3 mo. ALG, 12. 872.50 UNG eta dy 2% 
17. 324 6 mo. 8% 135-182 24 mo. 21% 
8. 800 90 d. Th% 14. 6000 4 mo. 41% 
9. 960 60 d. 4% 15. 4297 90 da. o% 
iA we Pal p un yesh (ck 319, 16. 845 6 mo. 7% 


313.— Bank Discount: When anote is discounted at date, the time 

to Find the Term named in the note (with grace or without, as 

Discount. the case may be) is the term of discount, 

showing how long the discounter’s money 

is used. A note, however, may be sold or discounted at any time 
between date and maturity. 


Norsr, — The method of reckoning the time from the day of discount to the day of maturity is 
not uniform among banks. 


The two more common methods are the following : — The tern yf discount 


extends from the day of 
discount to the day of 


I. When the time is less than two months, the exact 
number of days is counted ; but when it is more than two 
months, the time is reckoned in months and days as on 
p. 165. 

Il. The exact number of days is taken in all cases. 

In general, the latter method is used, when notes are 
large, being to the advantage of the bank. 

Thus, a 4-mo. note dated June 30 matures in Massachusetts Noy. 2. If discounted July 10, 
interest may be computed either for 115 days or for 8 mo. 23 da., the difference being $314 in favor 
of the bank on a $10,000 note. 

The student should conform to the custom of his own vicinity. Answers to examples are given 
for both methods. 


maturity. 


& 
Find date of maturity and term of discount by each method. 


Date of Note. Time Day of Date of Note. Time Day of 
to run, Discount. to run. Discount. 


May 8 60 d. June 10 Aprl4  4mo. June 8 

Nov.-l77 -90-¢. Dec. 14 Jan. 25 30d. Jan. 29 
Aug. 5 ) 1.2 mo: Aug. 29 Sept. 19 4 mo. Noy. 10 
Mar.17 3 mo. May 1 Feber” 7: 90 ar. Mar. 6 


ee a 


. 


en sande 8 en 


BANK DISCOUNT. 197 


314. — Bank Dis- 1. I get a 3-mo. note discounted 27 days 
count: Notes dis- after date. What is the term of discount ? 
counted after Date. 2. What if it were sold 24 d. before ma- 

turity ? 


3, A 6-mo. note for $450 dated Aug. 11 is discounted Sept. 4 
at 6%. KReckon months and days, with grace. Find proceeds. 

4. A 90-d. note for $1000 is discounted 37 d. before maturity. 
Find the proceeds, the rate being 5%. 


5. A d-mo. note for $800 was sold at 3% discount 80 days after 
it was made. Proceeds without grace ? 


6. A 90-d. note for $3000 dated Milwaukee, May 17, 1898, was 
purchased by the Traders’ Bank June 24. What did it yield the 
holder, money rating at 41 % ? 

7. 60 days; $450; 4%; date June 15; day of discount July 1. 


Y 


What does the borrower have for immediate use out of his note ? 


8. A 5-mo. note for $720 is sold in Trenton 7 days after its 
date at a discount of 9%. What does it yield? How much more 
or less would it have yielded in Chicago ? 


Using these data, find the proceeds of notes. Conform to the custom 
of your own vicinity in allowing grace and finding the term of discount. 


Date. Face. Time. When discounted. Rate. 

9. Aug. 21,’96 $800 90'da. Oct. 1 5% 
10. May 7,’97 $1218 4 mo. 31 d. after date 7% 

ll. Feb. 21, ’98 $249 60 d. Mar. 11 41, 

12. July 12,99 $1728 6 mo. 2 mo. 11d. bef. maturity 35% 
13. Nov. 16, ’95 $189.75 2 mo. Dec. 2 4% 
14. Dec. 20,97 $278  3mo. 164. after date 89, 
15. Aug. 17,’97 $350 72 d. At date 9% 
16. Sept. 12,’98 $760 5 mo. Oct. 22 3% 


17. Aug. 19,’99 $5000 60 da. Sept. 19 21% 


198 PROMISSORY NOTES. 


315. — Bank Dis- Notes that are discounted commonly bear 
count on Interest- nointerest. When an interest bearing note is 
bearing Notes. discounted, the maturity value of the note must 


be made the base of discount. 


PSOO___ Nfartford, Ct., tov. 12, 1895. 
Six monthe after date, 4 promise to pay to the 
order of Gdna Senet 
6ight /funcdred Dollars 
with tnteret at fowr her. cent, at the Gtna hank. 


Value received. 


Mary Warght. 


1. Is this note entitled to grace? 2 When does it mature ? 
38. How much must Mrs. Wright pay Mrs. French on that day to 
carry out her agreement? 4, Will transferring it to a third person 
affect the amount to be paid? £. What is the maturity value of 
the note ? 


6. If this note is discounted 4 mo. before maturity at a discount 
rate of 6%, what will be the proceeds? Explain the process. 


Process. 
102 %, of $800 = $816, maturity value; 98% of $816 = $799.68, proceeds. 


7. A 4-mo. note for $1200, drawing 9% interest, is discounted 
at 4% 3 months before maturity, without grace. Required the 
proceeds. 


8, Face, $500; rate of interest, 5%; time to run, 60 d.; term 
of discount, 48 d.; rate of discount, 3%. Proceeds? 
9, A 90-d. note for $720, dated May 15, 1896, drawing 8% 
interest, is discounted June 12, 1896. Proceeds ? 
10. Write an interest-bearing note, and find the proceeds. 


STOCKS. 199 


316. — Stocks. When a business requires more capital, or 
money invested, than is furnished by a few 
partners, a Stock Company or Corporation may be formed with any 
number of partners, who choose a Board of Directors to conduct the 
business as one person. 
The capital or stock furnished is then divided into equal shares of 
$ 100 each, unless otherwise stated, as $10, $50, $1000, ete. 


1, A corporation is formed with capital $250,000, shares each 
$100. How many shares are there? If there were only 500 
shares, each would be $ a. 

Each owner or Stockholder receives a Certificate of Stock bearing 
the seal of the corporation and giving the number and value of his 
shares. Shares are often sold, so that the shareholders are con- 
stantly changing. 


Incorporated under the Laws of the State of New York. 


No. 2/2. /2 wharew. 
EASTERN PACKING COMPANY. 
Chis certifies that Yultua /folm 
is entitled to - Twelve shares of One 
Hundred Dollars each of the full paid Capital Stock of 
the 


EASTERN PACKING COMPANY, 


Transferable only on the books of the company in person or by attorney 
upon surrender of this certificate. 


New York, Yuly /2, /895. 
Cart Yaecta, IH. FS. Kendatt, 


Secretary. President. 


2. rom whom does the Eastern Packing Company get the right 
to carry on business as a corporation ? 


200 PERCENTAGE. 


3. Who is the owner of the certificate showr at page 199 ? 

4,. What is the value of the shares represented ? 

5. A person owns 800 shares in a gas company valued at $50 
each. The total capital is $80,000. If each share has a vote, what 
part of the whole would he control ? 

The income or profit on the business is called a Dividend because 
divided and paid yearly, half-yearly, or quarterly, as a certain % on 
the par value or face value of each share. 

6. Compare the rate of income from a $1000 share paying 6% 
annually, and a $100 share paying 2% quarterly. 


317.— Stocks: The Market Value of stocks as quoted in the 
Market Value. newspapers varies from day to day. It is 
above par or below par accordingly as the 
dividends are large or small, and the business prosperous. The 
stock is then said to be at a premium or at a discount. 
Nore, — Look up stock quotations in some newspaper. 
1, If a share is quoted at 105, the premium is $a. It sells for 
2% above par. 
2. When a share sells for 15% below par, the stock is quoted at 
what price ? 
3. Find the cost of 25 $50 shares of mining stock offered at 10% 
discount. 


4, Sold 30 shares at 108; with the proceeds bought 10 at 112 
and 20 at 105. Balance = a. 


5. Bought insurance stock at 60% premium; paid $9600. How 
many shares did I get? 


6. If 17 shares cost $1725.50, one cost w, and the premium is y? 
7. What is the annual income on 37 shares paying 24% quarterly ? 


8. Suppose a person buys a share of bank stock at par. What 
does he pay? 9. If it yields 6% dividend, what rate of income does 
he get on his investment ? 


BONDS. 201 


10. Suppose he buys a share of railroad stock that yields 9%. If 
he pays the market price of $150, what rate of income will he get 
on his investment ? 

ll, If 23 shares are bought at 150, they cost « The dividend is 
6%. 78, x 23 x $100=y. yis what per cent of «? The rate of 
income is what ? 

15, Sometimes an Assessment is levied on the stockholders to pay 
the debts of the company. If the capital stock is $200,000, how 
much must be assessed on each share to pay a debt of $10,000 ? 
An owner of 40 shares would have to pay $2. 


318.— Bonds. When corporations, or national, state, and 

city governments borrow large sums of money, 

they usually give a series of Bonds, or promissory notes for one or 

more hundreds or thousands of dollars each, and to run several years 
at a fixed rate of interest. 

Whenever a payment of interest is due, it is usually collected by 
detaching from the bond a Coupon, or small certificate of the 
amount of interest then due, which is paid by the treasurer of the 
corporation or government. 

If interest and principal are not paid, the bondholders may law- 
fully take the property of those who gave the bonds. 


One of several interest coupons attached to a bond. 


1. If such a cou- 


THE NORTHERN LOAN ASSOCIATION pon is attached to a 


+3 Of St. Paul, Minnesota S- 7% bond, what is the 
Will pay to bearer at the office of | ace value of the 

bond ? 
2 When is the 
on the__/5___day of ___Uprt___1899,| next payment of in- 

being interest on cowpon bond No. /2/. terest due ? 

Edward fameo 3. Where can it 
: be collected, and by 

Secretary. 3 

whom ? 


the Company... Sauity five___ Dollars 


202 PERCENTAGE. 


4. If the bonds sell at 140, what would this one cost? 5. What 
would the annual rate of income be ? 


6. My 41% bonds yield me $180 annually. What is their par 
value? 7. But I get only 3% on what I paid for them, when they 
were quoted at x. 


8. Paid $1,770 for 16 $100-bonds and ordinary brokerage, 1% 
on par value. They were quoted at a. 


9. $9721.25 was the cost, including brokerage, of some bonds 
that were at a discount of 37. They were quoted at x What was 
the face of the bonds bought ? 


10. I bought 10 $500-bonds paying 4%, for $4912.50, including 
brokerage. The market value was a. 


11. A U.S. 420 bond (paying 4%, and maturing in 20 years) 
yields 3.478+ % on the investment. [Find the premium in dollars. 


319. — Stocks. As shown in the example on page 199, 
stocks and bonds may be registered on the 

books of the company or government issuing them, so that they 
cannot change owners except in writing and at the treasurer’s office. 


1. Under what circumstances may a stockholder get no dividend 
for several years? 2. How will this affect the value of the stock ? 
3. Why is it that the market price of bonds does not vary so much 
as that of stocks ? 


4, April 1, 1895, the U. 8. government had $ 100,000,000 of 5% 
bonds outstanding, payable Feb. 1, 1904. If these bonds are not 
paid till maturity, how much will the government pay out in interest 
on them ? 


5. When $ 50-shares of mining stock are quoted at $ 5, they are 
at a discount of a %. 6. When telephone stock sells at 230, it is at 
a premium of what per cent ? 


Trading in stocks and bonds is usually done through Brokers, as 
agents, who charge +% or more on the face value of these securities. 


PRESENT WORTH. 208 


7. When selling, does the brokerage increase or diminish the 
proceeds ? , How does it affect the cost when buying ? 


Find the cost, or proceeds, after paying brokerage. 


Number. Quoted at Brokerage. 
8, In buying 27 shares 105 1%. 
9, In buying 100 shares 7 1%. 
10, In selling 4 $ 1000-bonds 1004 1%. 
11. In selling 80 shares 538 25 ¢ each. 
12. In buying 8 $ 500-bonds 983 t%. 


13. If you know the amount of money invested, and the income 
received, how can you find the rate of income ? 


Apply the rule to the following examples : — 
14, 4% bonds at 104, and 4% brokerage. 
15, 15 shares at 91, paying 5%, no brokerage. 


320.— Present Worth 1, Henry Osmond owes me $ 600, due in 
of Non-interest-bearing a year without interest. We wish to find 
Debts due in the Future. what sum he shall pay me to-day, so that 

neither he nor I shall sustain loss; in 
other words, to find the present worth of a sum of money whose 
future worth is $ 600. 

2, What gives value to money? 38 At 6%, what is it worth to 
Mr. Osmond to have the use of $600 for a year? 4 The sum I 
receive from him now, plus the interest on it for a year, at 6%, 
should amount to how much? 6 Ought an arrangement whereby 
he has $36 at the end of the year and I have $600 to be mutually 
satisfactory ; and why ? 

6. Suppose, then, that he pays me $ 566.04, and retains $ 33.96. 
Find the amount of each of these sums at 6% for a year. What do 
I have at the end of the year? What does he have ? 

7, What, then, is the present worth of $600, due in a year, with- 
out interest ? 


204 INTEREST. 


To find the present worth of a non-interest-bearing debt of $ 600. 


Present worth Future worth 
$ 566.04 $ 600 
1 year 
$1.00 $ 1.06 
Process. 


_ The future worth of $1 ina year at 6% = $1.06; $600 is the future worth 
of as many dollars as $1.06 is contained times in $600 ; $600 + $1.06 = 566.04 ; 
hence the present worth of $600 due in a year without interest, is $566.04, and 
the true discount that should be made for the prepayment of the debt is $600 — 
$ 566.04, or, $33.96. 


Find the present worth. 
8. Debt $ 500, time 1 yr., rate 5%. 
9, Debt $ 750, time 2 yr., rate 4%. 
10. Debt $1200, time 21 yr., rate 6%. 
11. Debt $725, time 6 mo., rate 8%. 
12. Debt $ 86, time 4 mo., rate 9%. 
13. Debt $1100, time 5 yr., rate 10%. 


321. — Problems in 1, The interest of $600 for 4 yr. at 5% is 
Interest: the Product the product of what three numbers ? 
and Two Factors given 2. Make a problem in interest that shall 
to find the Third Factor. be represented by the equation, $84 =3 x 
7% X $ 400. 
8. Let 4 = interest, p = principal, r = rate %, ¢ = time in years, 
and show that i=» x 7 xt, or 2 = pri. 
4. Find the missing factor: 6 x 10 x a= 240; 180 =3 x & x 20. 


5. When the product and two of its factors are given, how is the 
third factor found ? 
7 Coho D 
pxr-? txr pxt 


6. When i= p x r X #, show that t= 


EXCHANGE, 205 


7. The principal is $400, the time 2 yr., the rate 5%. Explain 
the process of finding the interest, the time, the principal, and the 
rate by the use of the following formulas : — 


j=pxrxt=2x5%x$$400 =$§ 40; 
(= ¢ ner reretare aU = 2 yr.; 
pxr  5%x$400 20 7°? 
F AQ) 40 
I= SS eae = —_ = 400; 
hee 2x 5% iyo baie 
i 40 40» 
r= ——) = —_ = ° 
pxt 2 x 400 300°” 


8. In what time will $500 earn $ 60 at 6% ? 
9, At what rate will $ 400 earn $ 80 in 4 years ? 
10, What principal will earn $ 100 in 5 years at 10% ? 


Find the missing element. 


Principal. Interest. Time. Rate. 
id, $ 3000 $ 1080 | a 8% 
12. Ay 100.80 1 yr. 4 mo. 9% 
13, 780 87.75 2 yr. 5 mo. r 
14, a 16.00 60 da. 10% 
15, 480 A) 2 5% 
16, 1800 8.10 36 da. x 
Lf, 1725 w 2 yr. 11 mo. 19 da. 3L% 
322. — Exchange : 1. Mention some objections to sending 


Payment at a Distance coin or paper money by mail or express. 
without sending Money. 2. To sending United States money to 
Of Small Sums. Europe. 


The Postal Service and some Express Companies keep large sums in many 
offices. If you pay from 1% to $100 to a postmaster or express agent, he can 
write an order directing the pastmaster or agent at another office to pay the 
same sum to any person youname. For this accommodation you pay from 3 to 
30¢, no matter what the distance; or, if sending to a foreign country, from 
10 ¢ to $1, which is the cost of exchange. 

Nore. — Post-oflice orders are payable at an office named ; express orders, at any office of the 
same company, 


206 PERCENTAGE. 


3. Find the cost of money orders to pay the following debts. 


The charge for over $40 to $50 is 18 cents; over $50 to $60, 20 cents; over $60 to $70, 
25 cents. 

L. Ames of Oakland owes Ch. Adams of Newport $55; he owes 
J. Crane $ 69.90, and S. North $ 50. 


4. What must you pay in New York for an international money 
order for 106m. payable in Berlin, the rate being 10 % on each $ 10 
or fraction of $10 ? 


5. What would an order for 106 fr. cost? What part is cost of 
exchange ? 


323. — Exchange If a person, say Edw. Brown, keeps money 

of Any Sum by deposited at a national bank, or with a bank- 

Check. ing company, he may write a Check, as fol- 
lows : — 


New York, Xuly /2, 1895. 


THE WEST NATIONAL BANK OF NEW YORK. 
or order 


Dollars. 


No. &/. Gdward hrown. 


If indorsed in blank, —‘‘Simmons & Newton,’’ —it may be collected in 
New York by any person known to the West National Bank. 

If indorsed in full, — ‘* Pay to the order of James Gray, 

Simmons & Newton,’’ — 
it may be paid to James Gray or his order as soon as he has indorsed it. 

If sent to any city other than the one on which the check is drawn, it may 
be deposited in a bank and forwarded to the New York bank for collection, at 
a cost of 15 ¢ or more except to regular customers. 

Most debts are paid in this way. 


EXCHANGE. 207 


— 


1. Make a personal check for $3582.19, signed by A. Snow, pay- 
able to E. 8. Paine’s order. 2. Make a proper indorsement and 
explain the effect. 3. If the words “or order” were omitted, to 
whom would it be payable ? 

4. Is a check payable in more than one place? 5. If a Baltimore 
check is sent to pay a debt in St. Paul, does the creditor know at 
once that the debt is paid ? 


324. — Exchange : If a party, say Howe & Co. of Albany, 
of Any Sum by cannot draw a check, or if the creditor in 
Bank Draft. Syracuse will not accept one, a draft may 
be bought at a bank, for a small cost of 

exchange, like an order at a post-office. Thus: — 


B2SH___ Albany, Yuly AY Pete Fs) 
TENTH NATIONAL BANK. 
Pay to the order of /fowe ¥ Go. 
Two hundred thurly-fow 


Farmers’ National Bank, Satrtek Watthewe, 


Syracuse, N. Y. Cashier. 


1. Of what bank is Matthews cashier? 2. Where is the draft pay- 
able? 8. How can Howe & Co. make it payable to their creditors ? 


4. Make a draft from the National Exchange Bank of Peoria to 
the Traders’ Mutual Bank of Chicago. The Western Machine Co. 
pays $400 for it. 

5. Why is a bank draft more likely to be “ good” than a private 
check ? 


Nore. — A bank draft on New York may be cashed almost anywhere in the United States. Drafts 
on another city will usually be paid before collection in the region of which it is a commercial centre. 


208 PERCENTAGE. 


325. — Exchange: Money may be sent for, as well as sent. 
Collecting Debts by ‘Thus, if due to a Chicago firm from Haines 
Sight or Time Drafts. & Co. of Topeka, payment may be requested 

as follows : 


f/ 500% Chicago, July /2, 1895. 


At stoht [this makes it a sight draft] 
Sih wrly days af ter date [this makes it a time draft] 
Pay to the order o OQuiaever 
a. If. Wella 


Fifteen hundred and oo 


Value received, and charge to the account of 


; S. SF. Rieharda ¥ Co 
Jo. /fainew ¥ Co., Jopeka, Kaa. ; 


1. Read the above as a Sight Draft, then as a Time Draft, and 
explain the difference. 2. From whom is payment requested; that 
is, who is the drawee ? 3. Who is the creditor? The payee? The 
maker or drawer of the draft ? 

Such drafts, if known to be “good,’? may be sold to a bank by allowing a 


percentage for cost of exchange, or they may be forwarded through banks for 
collection. 


If the drawee on receiving a time draft agrees to pay it, he writes 
across the face of the draft the word “ Accepted,” followed by his 
signature and the date. This converts the draft into an acceptance 
and gives it precisely the force of the drawee’s promissory note, 
owned by the payee. 


4. Suppose the above, as a 30-day draft, to have been accepted, and 
offered for discount July 17. What is the day of maturity? 5. Find 
the discount at 6%, and the proceeds. 


FOREIGN EXCHANGE. . 209 


ce 


6. Suppose it had been bought by a bank the day it was drawn, 
and 3% charged for exchange besides the time discount. The pro- 
ceeds would have been «a. 


7. HF. Alto of New Orleans draws at sight on R. Fay of Waco, 
Tex., Aug. 3, 1895, for $ 500. Make the draft. 


8. Suppose the debt not due till Nov. 1. Make a proper time 
draft dated Aug. 3. 


9. If discounted Aug. 18, the proceeds would be a. 


10. If discounted Aug. 3, less an additional 1% for exchange, the 
proceeds would be y. 


11. Paine of Macon, Ga., owes Drew of Atlanta $4000, due Jan. 1. 
Paine accepts a draft Oct. 1 and discounts it himself for $a. ' 


12. A Charleston bank buys a draft on Richmond for $2100, 
charging 3% exchange. If it had taken the draft for collection only, 
the charge would have been 25¢. To the maker what is the differ- 
ence in money? In which case is payment made more quickly ? 


326. — Foreign Payment in foreign countries without send- 
Exchange. ing money is Foreign Exchange. Drafts are 
also called Bills of Exchange. 
1. Define Domestic Exchange, and mention several kinds. 
Bills of Exchange are drawn by one banker or broker on another. The rate 
of exchange varies from day to day, a pound sterling costing a little more or 


less than its bullion value ; 4 marks, a little more or less than $ 0.952; while $1 
will buy about 5.18 francs. 


2. Find the cost of a sight draft on London for £ 800, exchange 
at $ 4.90. 

3. A sixty-day bill can be bought at $4.87. What is the face 
of one that costs $9788.70 ? 

4. A debt of $ 4199.24 is due in London to a firm in Philadelphia. 


lor what amount sterling must a draft be made, if sold in Phila- 
delphia at $4.88 ? 


210 PERCENTAGE. 


5. Hind the face of a bill on Hamburg costing $ 871.525, exchange 
at $ 0.953 (= 4 m.). / 

6. What must be paid for a draft on Paris for 2000 francs, ex- 
change being quoted at fr. 5.16 (= $1)? 


327. — Review 1. (171)? — V 381 = 10% of what? 
Exercises. 2. How much pays this bill ? 
Written. ot yd. silk, at $2.50; 


2 pr. blankets, at $ 7.374; 

10% off to the trade and 2% for cash. 
8. Add horizontally : 3.75, 23.08, 176.97, 0.833, 12.374. 
4. $156.91, $73.99, $1439, $76.84, $974, $42.97, $1982. 


Find cost, but write only the products for adding : — 


t 162 at 80¢ 7. 901 at 25¢ 9 lat 20¢° 
48 at 662 ¢ 1200 at 75¢ 1,8, at 371 ¢ 
75 at 121 ¢ 12 at $ 1.162 43 at 25¢ 
24 at 624 39 at $ 2.334 100 at 27% 
374 at 10¢ 42 at 831¢ +b; at $ 1.25 
6. 6400 at 871 ¢ 8 22 at21¢ 10, 35; at 90 
279 at 111 ¢ 47 at 50¢ 48 at 183¢ 
108 at 81 ¢ si, at $ 1.00 1000 at 22 ¢ 
144 at 61¢ 103 at 28 ¢ ii, at $ 2.50 
1608 at 331 ¢ 161 at 331¢ 1,1, at 871 ¢ 


Find total interest due on jive notes, as follows. Write only results 
for adding. 


ae 12. 13. 
$500, 2tyr., 4%. $720, 45d, 6%. $700, 4yr., 43%. 
630, Fs ay 8%. 376, 30 d., 3%. 500, 90d., 6%. 
180, 30 i 12%. 200, 12. d., 4%. 400, 63d, 3%. 
900, 4mo., 9%. 820, 10 m.,-6%. 480, 2Lyr., 5%. 


200, 15d. 6%. 1800, 10 d., 4%. 150, 54d. 6% 


EXERCISES. 211 


14. Bought 500 tons Franklin coal, at $ 5.621. Sold at an average 
advance of 22%, but lost 5% in bad debts. Required net gain. 
Allow 25 ¢ a ton storage, and $32.60 for other expenses. 


Find proceeds of notes without grace : — 


16. WE 19. 
$ 60, 3 mos., 8%. $ 850, 47 d., 4%. $ 1900, 63 d., 4%. 
16. 18. 20. 


$ — 260, 904.6%. $946, 62d, 3%. $1217, 144, 8%. 


328. — Commission, 1. Estimate the commission on a sale of 
Insurance, $ 5000, at 5%. 
Taxes, Duties, etc. 


2. On an investment of $5000, less the 


Written. commission, at the same rate. 


3. Insured a mill, valued at a quarter of a 
million of dollars, for $200,000, at the following rates for 5 years: 
$ 50,000 in each of three companies, at ?% ; 320,000 at 2% ; and the 
remainder at %. What per cent of the value of the property is 
my annual premium ? 


4, $40,000 is to be raised by taxation for a schoolhouse. The 
assessed valuation of the town is $6,400,000. My property is 
assessed at $25,000. What shall I pay towards the cost of the 
schoolhouse ? 


§, An importation of 380 yd. of cloth, invoiced at 14 shillings, 
pays an ad valorem duty of 20%. It sells at $4 a yard. Gain on 
the lot? ($4.8665= ?) : 


6. Received $6000 for the purchase of wheat. Charged 2% 
commission, and paid 90 cents a bushel. How many bushels did 
I buy ? 


7, A lawyer collects a debt for me at a commission of $100 
guaranteed and 5%. He sends me a check for $3880. What was 
the amount of the debt ? 


912 PERCENTAGE. 


8. Imported 40,000 lb. sugar, invoiced at 23%, at 40% ad 
valorem, and 1¢ per pound, specific. Required the cost of the 
importation. 

9, Bought 19 shares of 7% manufacturing stock, at 1283. Re- 
ceived a semiannual dividend, and then sold for 126. No brokerage. 
My gain or loss ? 

10. Sold $50,000 worth of hides, at 2% commission, and with the 
net proceeds bought cotton at the same commission. Required my 
total commission. 


329.— Interest and 1, What is meant by principal ? interest ? 
Bank Discount. rate ? amount? proceeds ? term of discount ? 
Written. days of grace? bank discount? promissory 
note? indorser? indorsement? partial pay- 

ment? exact interest ? protest ? 

2. How large a check will pay a note of $385, that has been 
drawing 5% interest from June 10, ’95, to May 5, ’96? 

8. Bought an acre of land for $3500, and sold it at once for 10¢ 
a foot, taking a 4-mo. note, which I have discounted immediately at 
7%. My profits? Allow grace. 

4, Find the exact interest of $ 725, at 4%, for 125 days. 

5. A note for $1200, dated Aug. 19, 1895, bears 8% interest. It 
has one indorsement of $800, June 19, 1896. What will settle the 
note 3 years from date ? 

Find the missing term : — 

6. Principal, $800; time, 50 d.; rate, 3%; interest, $ a. 

7. Principal, $a; time, 4 yr.; rate, 10% ; interest, $ 328. 
8, Principal, $ 600; time, w yr.; rate, 4% ; interest, $ 60. 
9, Principal, $425; time, 3 mo.; rate, x %; interest, $ 95. 

10. What is the present worth of a non-interest-bearing debt, 
whose future worth in 8 mo. is $960? Money at 9%. 

ll, Face of 90-day note, $696; discounted with grace 25 days 
after date at 44%. : 


EQUAL RATIOS. 2138 


12, Required the proceeds of $500 note for 60 days, dated Dee. 
15, 1896, discounted June 12, 1897, at 5%. 


18. What may Charles French draw from a savings-bank paying 
4%, in semiannual installments; his deposit, $200 for 18 months ? 


14, Compare the interest, exact interest, compound interest, bank 
discount, and true discount of $600 for 1 year 4 months, at 6%. 


15. What amount will draw 3-mo. interest to Jan. 15? [$ 309. | 


Savings-bank Deposits. Withdrawals. 
June 12 $ 20 Aug. 18 $10 
Oct. 14 30 Oct. 16 20 
Oct. 17 15 WOVe ot 0 


330.— Proportion: [Review sections 92, 93; 146-149. ] 

an Equality of Ratios. 1. Define ratio; antecedent; consequent; 
couplet. 

2. 15+3=5; 43=5; 15:3=5. Give three different names to 
15, as used in these expressions; to 3; to 5. 

38. Will the inverse ratio, 5--15—=4, show the relative size of 
the numbers equally well ? 

4, What is the direct ratio of 12 and 36? The inverse ratio ? 

5. Compare the following ratios or fractions : — 

12:3 and 16:4; 6:20 and 6:24; 7; and i. 


Two equal ratios make a Proportion; thus : — 


12—16 5 —— 6 3 == 1 
| erey 8 Dare 2 A Sica 
4=4 ae ed 


Changing these ratios to fractional form, and then to smallest 
terms, we find them to be equal. The four numbers are therefore in 
proportion, and we say, for example, that 12 is to 3 as 16 to 4; or 
that the ratio of 5 to 20 is the same as the ratio of 6 to 24; or that 
3 bears the same relation to 18 that 1 does to 6. 


914 PROPORTION. 


Test the following proportions by seeing whether the ratios are equal: 


Ore 4 = OA ll; 2; ides 9 

ee o ee ee 2 LA A 2 Dero. 

af dae ea a 13. De 4% =121% : 25% 

9 21: 5=41: 9 14. 3yd.:4 yd. = $ 0.75: $1.00 


10, Seca 202 


The first and last terms of a proportion are the extremes; the 
second and third terms are the means. 

15. Which terms of a proportion are antecedents? Which are 
consequents ? Which are dividends? Which, divisors? Which 
may be numerators ? Which, denominators ? 

16. Arranging the proportion 16:8 =10:5 in a fractional form, 
we have 18 = 1°; multiplying both sides of the eae by 40, the 
l.c.m. of ie denominators, we have 40 x 16 = 40 x 1°; cancelling 
8 on one side and 5 on the other, 5 x 146=8 x 10; but 5 5 x 16 is the 
product of the extremes, and 8 x 10, the 
product of the means. What conclu- In a proportion, the 
sion may we draw regarding a_ pro- product of the extremes 
portion ? is equal to the product 

17. Test the proportions in Ex. 6-14 of the means. 
by this principle or law. 

18. Find the values of «:— 

EX Leese ee ae) Os Dine ey LS (ise eht 
Duet DEAL Dai bieel Oe 2 24:=— 61-12 

19. How is a missing factor found? How is the missing term of 

a proportion found ? 


Find the missing term in these proportions : — 


20; 17 :60= 852% 26. $2: $9 = 60 lb.: 27 lb. 
21. 27:a2=—81:100 av. 10T.: iT. = $a: $3.75 
Doe LO 150.2 26 tee J Bae PLS Y ts 

20 eoLd = 245 =? ee PACT a Rete AE eee high 

24. =,:2=8:16 380. 94:161 = 38:2 


2b.. $90: $4816 yd. se yd. eh) 1.25: 0.5475 


RULE OF THREE. 915 


331. — Proportion: Proportion may be apphed to the solution of 
the Rule of Three. problems in which three terms are given to 
find a fourth. 
Of the three terms given, two are like numbers and the third is 
like the required result. 
1. Why must the terms of a ratio be like numbers ? 
A. 16 yd.:2 yd. = $32: $4 B. 3:18= $6: $36 
2. In A which ratio is larger than1? 3. Howisitin B? 
4. Show that the ratios of a proportion must correspond, both 
being larger than 1, or both smaller than 1. 
5. If 16 yards cost $40, what will 10 yards cost ? 


Process. 6. What is the denomination of the 
16:10 = $40: $2 or $25, answer, or 4th term of the proportion ? 
5 5 7. Why must the 3d term be dollars 
1p x BAD _ ® 25 also ? 
16 8. Will 10 yards cost more or less 
than 16 yards? 9. Then will the 


2d ratio be greater or less than 1? 10. If the ratios of a pro- 
portion must correspond, must the 1st ratio be made greater or less 
than 1? 


11. Read the incomplete proportion. 12. How is the missing 
extreme found ? 


18. What is the ratio of 16to 10? Of $40 to $25? 14. Perform 
the problem by analysis. | 


Solve the following by the “ Rule of Three” : — 


15. When 60 bushels of oats cost $36, what will 25 bushels cost ? 

16. What will 18 tons of hay cost if 7 tons cost $147 ? 

17. If 9 weeks’ board costs $ 94.50, what will 12 weeks’ board cost ? 

18. If a yacht sails 24 miles in 70 min., how long will she be in 
sailing 108 miles ? 

19. If 72 men lay 2 miles of water pipe in 15 days, how many 
days will 48 men require ? 


216 PROPORTION. 


20. If a train runs 1000 miles in 28 hours, how many, miles can it 
run in 120 hours ? 

21. If 2% yards of cloth can be bought for $ 23.10, what should be 
paid for 152 yards at the same rate ? 

22. If 1 acre yields 22 bu. 3 pk. of corn, how many acres would 
yield 546 bu. ? 

23. If a 5-cent loaf of bread weighs 8 ounces when flour is worth 
$5, what would be the proportional weight when flour is at $6? 

24. 7 of a yard costs $2; at the same rate 3 yard costs $ a. 


332.— Partnership. 1. James Smith and Charles Butler form 

a partnership under the firm name of James 

Smith & Co., with a capital of $5000, of which Smith furnishes 

$4000 and Butler $1000. What part of the capital does each 
furnish ? 

2. They agree to share gains and losses in proportion to the cap- 
ital each furnishes. They gain $1500 the first year. How shall it 
be divided between them ? 

3. The second year they lose $700. What should each pay ? 

4. In the absence of an agreement between the partners, is it just 
to divide the gain according to the capital furnished ? 

5. Divide $9000 profit among three partners who furnish 
$ 20,000, $380,000, and $40,000, respectively. 

6. The profits were $4800. Dunlap’s share was $2000. What 
part of the capital did he put in ? 

7. Two young men hire a bicycle for $4 a week. One has it 
Mondays and Thursdays. What is his share of the expense ? 
The bicycle rests on the Sabbath. 

8. Divide $750 between a girl and her brother in proportion to 
their ages of 6 and 9. (Their united ages are the capital.) 

9. A furnished the store, and B the stock, worth $6000. Of the 
profits B had 87} cents out of every dollar. What was the store 
worth ? 


SQUARE ROOT. O17 


— 


10. A bankrupt firm’s assets are $27,500, and their liabilities 
$32,500. What do they pay on a dollar? They owe me $600. 
I get $a. 

11. A’s gains are 43 of the total gains. B’s capital is $8400. 
What is A’s capital ? 

12. Divide $15,000 among 3 men who furnished capital at the rate 
of $4, $4, and $4 each. 

13. Two partners earn 10% on their capital besides the $ 2000 
which each takes in weekly allowances. One receives in all $ 3500, 
the other $ 2500. . What is the total capital ? 


333.— Powers ana 1. Find these powers and roots: 
Roots: how a Number 107; 100; 40°; 1600; 70?; 4900 
is Squared. 202; 400; 50?; 2500; 802; 6400 
30°; V900; 602; 3600; 90?; 8100 
2. How is anumber squared? 8. How are roots related to pow- 
ers? 4. How many equal factors make a square? <A cube ? 


Extracting the square root of a number, or separating it into two 
equal factors, is the reverse of squaring one of the equal factors. 
A careful analysis of the process of squaring a number will enable 
us to reverse the process and find the square root of a number. 


A In A we first multiply by the ones as usual. B 
47 In B we begin with the tens. 40+7=47 
a Nore, —¢=tens; 0=ones; 240 =2 x tens X ones. 40-7 = 47 
ioe txt = OneR t2 = 40 x 40 = 1600 
280 = 7 x 40 = ones x tens -xXO= 40x F260) 
280 = 40 x 7 =tens x ones Oe be 7156.40 =" 280 
1600 = 40 x 40 = tens? a Ma Sa 
2209 = 47 x 47 = tens? + /tens x \+/ones x \ + ones? = # + 2to + 0? = 2209 
( ones ( tens 


5. In these processes of squaring 47, how many partial products 
are obtained? 6. What is the largest? The smallest? 7 Com- 
pare the remaining two and show how each is found. 


VARS) POWERS AND ROOTS. 


8. What is the difference between 7 x 40 


C and 40 x 7? 9. What is the square of the 
Fu = me fet tens? The square of the ones? 10. The 
1600 =402 =? product of the tens and the ones? 11. What 
560 =2x (40x 7) =2to 18 the product of the tens and ones taken 
49 = 72 = 0? twice, as in C’? 


2209 = ? + 210 + 0? 12. Show that the square of 47 contains 


the square of its tens, the square of its ones, 
and twice their product. 


56 18. Square 56, and show what three parts its 
ay square contains. 
20044. = arte ls 
600 = 2to0 = Part II. 
36= 02 = Parti. 933 96. 


3136 15. Without writing the process, square 24; 
BOs Hoss OG Ton 


14. In the same way square 45; 28; 32; 44; 


16. Explain what three The square of a number of two 
parts the square of every | digits contains — 
number of two digits con- The square of the tens, the square of 
tains. the ones, and twice their product. 


334. — Squares 1. Square 10 and 99 and any number be- 
and their Roots com- tween them. Compare the number of places 
pared as to the Num- in the root with the number in the power. 


es Places, they 2. Do the same with 100 and 999, and any 
BORON other number of three figures. 
3. How many figures in 1000°? In 10000? ? 


4. If the root has three figures, the square will have how many ? 
If the square has eight figures, how many has the root ? 


5. How many figures in the square of 33? 75? 207? 964? 


28796? 6. How many figures in the square root of 9409? 381? 
7225? 182329? 49434961 ? 


SQUARE ROOT. 919 


7. How does the number of places in a square compare with the 
number of places in its root ? 
8. Square 0.2; 0.02; 0.4; 0.12; 0.25; 0.03; 0.005. 
9. Compare the number of decimal places in power and root. 
10. Why can the square of a decimal never contain an odd number 
of decimal places? 11. Which is larger, 3 or (3)?? 18 or V48? A 
fraction or its root ? 


335. — Extracting 1. Write the squares of 10, 20, 30, and so 
Square Roots: find- on to that of 100. 


ing two Equal Factors 2. Write the squares of hundreds, 100, 200, 
of a Number. 300, ete., to that of 1000. 


To find the square root, or one of the two equal factors, of 2809. 


Taceay. 38. How many figures in this square ? 

2 +2 to + 0? = 2809 (50 +3 In its root? 4. Of what three parts 

#2 or 502 = 2500 = Part I, does the square consist? 5. What is 

2tor100) 309=2t0+o02 thesquareof 50? Of 60? 6. Between 

2to or 100 x 3= 300 = Part U. what two squares does 2809 come? 

o2or3?= 9=Part Ill. 7. Then its root is between what two 
numbers ? 


8. What is @, or Part I. of the square? 9. Taking it out of the 


square, what two parts remain? 10. What are the tens of the root? 
Twice the tens, or 2t? 


ll. 3x «=24. How isa missing factor found? 12. Assuming 
309 to be the product of two factors, 2¢ and o, and calling 2¢ 100, 
what is 0, or the ones of the root? 18. If 2¢= 100 and 0 = 3, what 
is 2to, or Part II. of the root ? 

14. Taking 300, Part II. of the root, out of 309, what remains. 
15. What part of the square is this? 16. What, then, is the square 
root of 2809? Prove it. 

In the same way jfind— 

17. V529 +18. V676 19. V1156 20. V1764 21. 2025 
22. 2916 28. V3969 24. V4624 25. 5625 26. 7056 


220) POWERS AND ROOTS. 


A Shortened Process. We may shorten the process by omit- 
+ 2to+0?= 88/36 (94 ting ciphers both in # and in dividing 
gigeasler > by 2t. In 180)736, show that — 
2¢=180| 7386 =(2t+0) x o (2Qt+0)xo=2to+0? 
pees or (180 + 4) x 4 = 720+ 16 or 736. 


2t+0 = 184 736 = 2 to + 0? 2YSE 
| Solve by both processes. — 27. 4096 


28. 5329 29. 6889 380. V8464 31. 3364 82 /9801 


336. — Square The process on the preceding page may 
Root of Large Num-___ be applied in finding the root of any number. 
bers, Decimals, and 

To find the square root o Pause 
Fractions. fi d J 601.7209 


Explain each step of the process, telling 
how we get each number. 


Process. SuacEstions.— We begin at the point 
6/01'.72/09(24.53 and separate the power into 2-figure groups, 
4 showing that the root has 4 figures. We 
2t—40; 44 | 201 first use the left-hand groups, 6/01, to find 
176 the first figures of the root, 24. We then 
2t=480; 485| 2572 annex the third group, 72, and treat the 24 
2425 as the tens of the root, and soon. Having 
2t = 4900; 4903} 14709 found the third root-figure, 5, we consider 
14709 245 as the tens of the root, etc. 
1. 283024 8. V/404496 6. V755161 
2. ~/299209 4. 556516 6. 2137444 


To find the root of 0.501. 


ExpLanation.— We begin as before at the point 


Process. and separate the power into 2-figure groups, annexing a 
0.50'10(0.707+ zero to fill the second group. As no decimal power can 

49 have a partial group, we know that this decimal is an 

1407 | 1 1000 imperfect power. For the third root-figure we annex 
9849 a cipher-group, and proceed as before, using + or — to 

1151 mark an approximate root. The work might have been 


carried farther, 


24. 


my ob Abh 


OF FRACTIONS. 


11. 0.89 
12. 19.467 


18. 824.9 
14. 17.035 


15. 


V 0.64 
. V0.064 
. V19382.4 


In 
fractions: 


19. 
20. 
21. 
22. 


AN | 
V/ 2044900 
\/76.3876 
V0.8 


finding the root of 


I. First change them to 
simplest form, as in A or C. 
Il. Use the method in A 


or 


perfect powers. 


Ill. Use B or E when both terms are imperfect powers. 


ve 


25. V6qg 
26. V5R 


337.— Extract the Square Root. 


Oral. 
1. V14400 ~— 11. -V/0.49 
2. 48 12. 0.049 
8. (54) 13. V0.00490 
4. (161) 14. V625 
5. V0.09 15. 0.625 
6. V36 x 49 16. V16 million 
Foavelico “all. tt 
8. (875) =—-:18. V304 
9. 3V81 19. 2724 
10. 0.0625 20. V10.% 


OT. \/824 
28. \/1905 
Written. 

1. V94249 iB 
2. V0.729 12. 
8. V137 13. 
4. /1008016 ~=«14. 
5. \/9834496 15. 
6. 62742241. 16. 
7. \/2033.1081 17 
8. 39 18 
9. V8+4+5+8 19 
10. 998001 20 


D may be used when the denominator is a square. 


C when both terms are 


29. V/1514 
80. 2,8, 


. V3444736 
. V1 +25? 
. V0.741 


229 MENSURATION, 


338.—To find Any 1. Draw aright triangle with base 14 inches 
Side of a Right long and perpendicular 2 inches. 
Triangle. 2. On each of the three sides as base draw 
a square. 38. Separate each square into half- 
inch squares. 4. Compare the squares on the hypotenuse with the 
sum of the squares on the other two sides. 

5. If from the square on the hypotenuse you take the square on 
the base the remaining area will equal what square ? 

6. The square of the base is 56; the 
square of the hypotenuse is 100; the 
square of the perpendicular is a. 
Prove this by drawing a triangle with 
_ squares on its sides. 

7. Hypotenuse? = 225 


In a right triangle, 
The square of the hypot- 
enuse equals the sum of the 


squares of the other two 
sides. 


Perpendicular? = 144 8). 3? 447 256 ee aoe. 
Base =a 9. A? = 6253, B= 400 ea. 
10. Explain these formulas: 11. The three sides of a right 

H=V BaP? triangle are respectively 39 in., 


65 in.,and 42 in. With any two 
given find the other. 


B.=vV H?— Pp 
P=V' He 3B 
Explain the following process: 
H=V B+ P? = Vv 42? + 39 = V1764 + 1521 = V4225 = 65 
B= WV H?— P? = Vv 65? — 422 = V 4225 — 1764 = V 1521 = 39 
P= V H?— B= V 65? — 892 = V 4225 — 1521 = V 1764 = 42 
Find the unknown sides of the right triangle, drawing a figure and 
marking the dimensions in each case : — 


Hypotenuse. Base.  Perpendic. Hi. B. P. 
12. 55 x 33 Lis 162 70 x 
13 26 14 x 18. x 39 27 
14 36 20 av 19. 208 x 93 
15. ev 15 60 20. Ly, 13 


16. 325 2 18 21. lope 


rho 


APPLICATIONS OF SQUARE ROOT, 993 


339. — Practical 1, The top of a square table has an area of 
Application of 576 sq. in. What is its length ? 
Square Root. 2. What is the length of a square field con- 
Written. taining 10 acres? Its perimeter ? 


3, A rectangle measures 22 ft. by 10 ft. 

How long is its diagonal ? 

4, The foot of a 25 ft. ladder is 12 ft. from the side of the house 
against which it leans. How far from the ground 1s its top ? 

5. What is the area of a right triangle whose longest side is 20 
ft., and its shortest 8 ft. ? 

6. Find the diagonal of a 36-inch square. 

7. Find the altitude of an equilateral triangle whose side meas- 
ures 24 ft. 

8. What will it cost to fence a square field containing 5 A., at 
$ 1.25 a rod? 

9, A pitch-roof house is 22 ft. wide. The ridge-pole is 10 ft. 
higher than the plate. How long are the rafters if they project 1 ft. ? 


10, Find the area of an isosceles triangle whose base is 12 ft., and 
its perimeter 50 ft. | 


340. Rectangles and 1, What is the length of a square equal 
Triangles. in area to a rectangle 24 rd. long and 33 ft. 
Written. wide ? 
2. What is the longest straight line that 
can be drawn on the ceiling of your schoolroom if it measures 32 
ft. by 30 ft. ? 
3. Compare the perimeter of a rectangle 48 in. by 12 in. with 
that of a square of equal area. 
4, How much do I save by crossing along its diagonal a square 
that contains 1296 sq. rods instead of going round its two sides ? 


5. How long is an acre of land in the form of a square ? 


224 MENSURATION. 


6. How long a guy will support a derrick 48 ft. high if fastened 
85 ft. from its base ? | 

7. The hypotenuse of a right triangle measures 90 ft. The other 
sides are equal. How long are they ? 

8. What is the shortest possible distance that I must walk to go 
from the center of a 10-acre square field to each corner, and return 
to the starting-point ? 


9. The area of a circle= D? x 0.7854; then D= ee and 
, 0.7854 
D= Area What must be the diameter of a circle to contain 
0.7854 


approxunately an acre? 
10. Two poles are 100 ft. apart. One is 60 ft. high, and the other 
80 ft. How long a line will connect their tops ? 


341.— Contents of [Review section 249. ] 

Cones. 1. What part of a square prism is a cylin- 

der of equal diameter and height? 2. A 

square prism contains 10,000 cu. in. <A cylinder of the same 
diameter and altitude contains 0.7854 as much, or & cu. in. 

A Cone is a solid, 
having a circle for 
its base, tapering uni- 
formly to a point, the 
vertex of the cone. 
AC is the slant height. 


3. Construct a hol- 
— low cylinder and a 
" cone of equal base 
and altitude. Using sand or water, show that the cone is 4 of the 
cylinder. 
A circle = 0.7854 of a square of the same diameter. 
A cylinder = 0.7854 of a square prism of the same diameter and 
altitude. 


—| ———— 


Hh EL Ny 
HT Bi 


CONES AND PYRAMIDS. 295 


A cone = 4 of a cylinder or 4 of 0.7854, — that is, 0.2618, — of a 
square prism of equal base and height. 

4. Find the contents of a prism 4 in. square and 12 in. in altitude, 
and of the largest cylinder and cone that can be turned from it. 
Explain the process. 

Process. 12+4?'=192 = contents of square prism. 

0.7854 of 12 x 4° = 150.7968 = contents of cylinder. 
0.2618 of 12 x 42= 50.2656, or + of 150.7968 = contents of cone. 
5. Diam. of base = 15; alt. = 20. 
Find the contents of cones : 6.. Diam? of hase =. +8; alt; ==40. 
7. Diam. of base = 10; alt. = 36. 

8. First find the altitude and then the contents of a cone, the 
diameter of the base being 8, slant height 12. 


A Cone is 4 of a cylinder of 


equal base and altitude. 


342.— Contents of A Pyramid is a solid whose base is a regular 
Pyramids. polygon, and whose sides are triangles meet- 
ing im a common point, the 
vertex of the pyramid. 
AC is the slant height, 
Pyramids, like their bases, 
are square, triangular, hexag- 
onal, ete. 


A Pyramid is 4 of a prism 
of equal base and altitude. | 


1. A square pyramid 12 ft. high 
measures 3 ft. along one side of its base. Required its contents. 


296 MENSURATION, 


2. A granite shaft 10 feet high and 20 inches square is surmounted 
by a square pyramid 2 feet in altitude. The contents of both? 

3. Which is more easily measured, the slant height or the alti- 
tude of a pyramid? Which line of a triangle is the slant height of 
a pyramid ? 

4. The slant height of a square pyramid is 15 inches, and the 
diameter of the base 10 inches. Find its contents. 

5. The area of the base of a pentagonal prism is 621 Sq. 1n.; its 
altitude is 24 in.; the contents ? 


343.— Of the Con- 1. Of what form are the sides of a pyra- 
vex Surface of Pyra- mid? 2. How is the area of a triangle 
mids and Cones. found ? 

3. Find the convex or lateral surface of a 
square pyramid 16 in. in slant height, with base 1 ft. square. 

4. Altitude of a square pyramid = 20 in.; diameter of base = 10 
in. Lateral surface = a. 

5. Slant height of octagonal pyramid 24 in.; side of base 6 in.; 
lateral surface ? 

6. With a radius of 4 inches, draw any sec- 
tor, as A, B, or C, on thick paper. Cut it out, 
and place its radii together to form a hollow 
cone. 7. What line of the cone is the radius of 
the sector? The arc of the sector ? 

8. The product of the circumference of a 
circle and half its radius equals what? 9. The 
area of a sector is found like the area of a tri- 
angle. Explain from the figures that base x 
4 the altitude, or arc x 4 the radius = area. 

10. The circumference of the base of a cone 
(are of sector) is 21 inches; the slant height of 
the cone (radius of sector) is 10 inches. What 
is the area of its convex surface ? 

11. Circumference of base 60 inches, slant 
height 124 inches. Convex surface ? 


180 


SPHERES. 997 


12. The diameter of the base of a cone is 10 inches, its slant 
height is 15 inches. Required 
the convex surface. The convex surface of a 

13. How many square inches in cone or pyramid equals the 
the entire surface of acone 8 inches | product of the circumference of 
in diameter at the base, and 20 its base and 4 its slant height. 
inches in slant height ? 

14. How many square inches in the entire surface of a cone whose 
base is 10 inches and whose altitude is 24 inches ? 

16. What is obtained when the perimeter of the base of a cone or 
of a pyramid is multiplied by $ its slant height ? 

16. If the largest possible cone should be turned out of a square 
pyramid, what decimal part would become shavings ? 


344. — Surface 1. Cut away any slice of a sphere, as an 

of Spheres. apple. What is the form thus exposed ? 

2. When a sphere is bisected, the plane 
surfaces thus exposed are great circles of the sphere. Would the 
diameter and circumference of one of these circles be the diameter 
and circumference of the sphere ? 

3. It can be 
proved that the 
flat surface of a 
hemisphere is 4 of 
its curved surface. 
How many great 
circles in_ the 
curved surface of 
a hemisphere ? 

4. How many great circles in the surface of a sphere ? 

5. How is the area of a circle found ? 


If D? x 0.7854 = Area of a great circle of a sphere, 
then D? x 0.7854 x 4 = Area of 4 great circles, or of the sphere; 
but D? x 0.7854 x 4 = D® x 3.1416 = Area of sphere. 


98 MENSURATION. 
6. What is the area of the surface 
of a sphere 5 inches in diameter ? 
Explain. 
5? x 3.1416 = 78.54 square inches. 
7. How many square inches in the surface of a 12-inch globe ? 
8. Calling the diameter of the moon 2000 miles, how many square 
miles in its surface ? 
D? x 3.1416 = surface of sphere. 
But TF Xb A416 = Dex"( Dx A416), 
and (D x 3.1416) = the circumference. 
Hence D x (D x 3.1416) = D x circumference, and 
area of sphere = diameter x circumference. 
9. Diameter of sphere =7 inches. Its circumference = 22 inches. 


The surface of a sphere 
is 3.1416 times the square 


of its diameter. 


What is its area ? 


10. Diameter of earth = 8000 miles. Its circumference 25,000 
miles. Area of its surface = x miles. 


345. — Contents of 1. If a sphere should be dissected as in 
Spheres. the accompanying illustration, of what would 
it appear to be composed ? 


2. What line in the sphere forms the altitude of each pyramid ? 
3. What forms the base of each ? 


SIMILAR FIGURES. 229 


4, Taken together, what will the bases of all the pyramids make ? 

5. How are the contents of any one pyramid found ? 

6. What should we obtain by multiplying the surface of the 
sphere (the bases of all the pyramids) by 4 of its radius (the altitude 
of each of the pyramids) ? 

7. The surface of a sphere is 113 sq. in., and its radius 3 in. 
What are its contents ? 

8. How is the surface of a sphere found? + of the radius is 
what part of the diameter ? 

9. Read and explain the following: 

R 
= x D’x 3.1416 = contents of sphere. By cancelling, we obtain 


R_D 


x D’ x 3.1416 = volume of sphere; but oe hence 


© 
e 


Dx D’ x 0.5236, or D® x 0.5236, = Contents of Sphere. 
10. The diameter of a sphere is 24 
inches. Its contents ? 


The contents of a sphere 
11. How many cubic miles in the | are équal to 0.5236 of the 
moon, if of 2000 m. diameter ? cube of its diameter. 
12. If the largest possible sphere is 
turned out of a cube, what decimal part becomes shavings and 
what part sphere ? 
To be remembered : 
x = 3.1416, used in finding circumferences. 
1 of x = 0.7854, used in finding { ie PES 
1 of x = 0.5236, used in finding contents of sphere. ° 


iy of + = 0.2618, used in finding contents of cone. 


pe 0.31851, used in finding diameters. 


y 


346. — Comparison Similar surfaces have the same form. 
of Similar Surfaces. 1. Are all circles similar surfaces? Are 
all triangles? Are all equilateral triangles ? 
Mention other similar surfaces of different sizes. 


930 MENSURATION. 


2. Give values to x and y. 
652% (103: 
s fe O° aay eee, 


=e 3. In the figures at the left, what 
2 is the ratio of the side of B to the 
A 


side of A? Of the area of B to the 
area of A? 
4. Compare the side of C with 
e the side of A. Compare their 
areas. 
er 5. What is the ratio of H to D? 
Of their diameters ? 
6. Compare D and F, as to diame- 
ters and as to areas. 


Similar surfaces have 

7. How much more water will pass | the same ratio as the 
through a 2-inch nozzle than through a | squares of their corre- 
1-inch nozzle ? sponding lines. 


8. A faucet with a 4} in. opening 
will require 4 times as long to fill a tank as one with an inch open- 
ing. Explain. 

9. If a square lot 60 ft. long costs $300, what will one of the 
same shape 3 times as long cost ? 

10. Compare the strength of a rope 2 in. round with that of one 
3 in. round. 

ll. If it costs $8.25 to gild a sphere 20 inches in circumference, 
what will it cost to gild one of 30 inches ? 

12. A 2 in. faucet fills a tank in 28 min.; a 3 in. faucet in @ min. 

13. The ratio of two similar triangles is 25; the ratio of their 
altitudes is a. 

14. To paint a conical steeple 30 ft. high costs $35; to paint 
one 45 ft. high costs, at the same rate, $ a. 

15. A conical tent, 16 ft. slant height, costs $13.50; one measuring 
4 ft. more would cost $ a. 


SIMILAR SOLIDS. 921 


a 


347.— Comparison 1. Similar solids have the same form. Men- 
of Similar Solids. tion several similar solids. 


2. Compare the two corresponding edges in A and B. In Band 
C. In A and C. 


3. Compare a face of A with one of B. With one of C. 


4. Read and explain the following proportions : — 


Edge of A: edge of B= 2: 4. 


Edge of B: edge of C= 4:8. Similar solids have 
Surface of A: surface of B = 2?: 4”. the same ratio as the 
Surface of C’: surface of B= 8?: 4’. cubes of their corre- 
Volume of A: volume of B= 2°: 4°. sponding dimensions» 


Volume of B: volume of C= 4°: 8*. 

5. C will weigh how many times as much as A? As BY 

6. If a 2-inch sphere weighs 1 1b., how much will a 6-inch sphere 
weigh ? 

7. If a rectangular bin, 5 ft. long, contains 75 bu. of oats, how 
many bushels will a similar bin, 124 ft. long, contain ? 


8. It requires 90 min. to fill a cylindrical tank, 3} ft. in diameter. 
At the same rate how many minutes will be required to fill a similar 
tank, 14 ft. in diameter ? 


9. If a coil of +-in. wire weighs 48 lb., what will a coil of similar 
wire, 33,-in. in diameter, weigh ? 


10. Compare a }-in. cube with a cubic yard. 


Ay MENSURATION. 


348. — Mensuration. 1. Find the cost of plastering the walls 
Written. and ceiling of a room 18 ft. long, 15 ft. wide, 
and 9 ft. high, allowing 4 of the area of the 
walls for openings and wood-covered portions. Price 121¢ per 
square yard. 
2. Find the area of a right triangle, base 25 ft., hypotenuse 
60 ft. 


3. A rhomboidal field contains 5 acres, and measures 50 rods 
along a straight road. How wide is it? 


4. Name the six quadrilaterals and the four parallelograms. 
Draw a trapezoid and its equivalent rhomboid and rectangle. 


5. Required the contents of the surface of a stick of timber 18 
ft. long, 4 in. thick, 8 in. wide at one end, 12 in. at the other. 


6. The diagonal of a trapezium is 22} ft., and the perpendiculars 
drawn from the vertices of the angles opposite it are 16 ft. and 12 
ft., respectively. What is the area of the trapezium? Draw it. 

7. Find the cost of 28 six-by-four joists, averaging 18 ft. in 
length, at $52 per M. 

8. What is the area of a walk 3 ft. wide, around a semicircular 
flower-bed, the straight edge of which measures 12 ft. ? 

9. There is a difference of 6 in. in the diameter of the wheels of 
a carriage. The fore-wheel turns 1000 times in going a certain dis- 
tance. The hind-wheel turns a times and is 4 ft. in diameter. 

10. What is the axis of a sphere 4 ft. in circumference ? 

11. From a sheet of zinc, weighing 16 lb., and measuring 8 ft. by 
4 ft., a square was cut, reducing its weight to 114 lb. How long 
was the square ? 

12. How much ground is covered by 12 cords of 4-ft. wood piled 
2 ft. high ? 

18. A granite sphere barely clears a gateway 2 feet wide. Find 
its contents. 

14. Three semicircles are so arranged that their 3 ft. diameters 
enclose a triangle. Find the area of the whole figure. 


EXERCISES. 233 


349.— Mensuration. 1. A span of horses draws a load of brick 
Written. weighing two tons. The brick are of the 
ordinary size, 8 x 4 x 2, and weigh 100 lb. 

to the cubic foot. How many bricks in the load ? 

2. Find the contents of a cone 6 in. in altitude and 2 in. in 
diameter at the base. 

3. Find the slant height of a square prism 12 in. in altitude, base 
8 in. on a side. 

4. Find the entire area of a hemisphere 15 in. in diameter. 

5. How long is the equator, the equatorial semi-diameter of the 
earth being 3963.296 miles ? 

6. Compare the perimeters of a rectangular field 60 rd. by 30 
rd., of an equivalent square field, and of a circular field of the same 
area. 

7. The inside dimensions of a cellar are 16 ft., 12 ft., and 8 ft. 
The wall is to be two feet thick. How many cubic yards of earth 
will need to be removed ? 

8. A cylindrical 8-inch driven well is 76 ft. deep. How many 
cubic feet of earth, etc., have been taken out ? : 

9. One fire engine throws a 2-inch stream, another a 13-inch 
stream. Compare the amount of water thrown. 

10. One piece of shafting 2 in. in diameter weighs 500 lb. What 
is the weight of a similar piece 31 in. in diameter ? 

11. How deep shall a 12-ft.-square bin be made to hold 1728 
bushels ? 

12. How many cords of wood in a section of a giant pine 18 ft. 
long and 16 ft. in diameter ? 

18. If a 2-inch rope breaks with a weight of 8000 lb., what 
weight might break a similar rope 3 in. in circumference ? 

14. Give area of basin required for a fountain that throws its 
spray out 15 feet. 

15. A cone of cinder 50 ft. high is 314.16 ft. in circumference at 
the base. What is the shortest line of ascent ? 

16. A park 1 m. square is bounded by a 10-ft. walk inside the 
wall. How many squares of cement cover the walk ? 


234 


DEFINITIONS. 


350. — DEFINITIONS. 


[FOR REFERENCE. ] 


Acceptance. The formal agree- 
ment by signature of a drawee to pay 
a draft according to its terms. 


Agent or Correspondent. One 
employed to transact business for 
another. 


Assessment. Money collected 
from shareholders in stock companies 
to meet losses or expenses. 


Assessors. Officers who estimate 
the value of taxable property and 
apportion the tax to be raised. 


Average of Accounts. ‘The proc- 
ess of finding the equated time for 
the payment of the balance of an 
account. 


Bank. A corporation formed to 
trade in money and securities, or for 
the custody and loaning of money. 


Bank Discount. The allowance 
made to a bank by the holder of a 
note for having it paid to him before 
maturity. 

Base. The number of which a 
percentage is taken. 


Bill of Exchange. A general term 
for foreign or domestic drafts, espe- 
cially for the former. 


Bonds. A series of interest-bearing 
notes of a government or corporation. 


Broker. An agent who buys and 
sells securities or other property. 


Brokerage. A broker’s fee or com- 
4 


mission. 


Capital. Money or other property 
invested in business. 
Charter. A special act of a legis- 


lature setting forth the rights and 
duties of a corporation. 


Check. A depositor’s order for the 
payment of money by his bank. 


Commission. A percentage paid 
to an agent for transacting business 
for another. 


Compound Ratio. The indicated 
product of two or more simple ratios, 


as 3:2x 4:8, or0 72h. 
4:8 


Cone. A solid having a circle for 
its base, and tapering uniformly to a 
point, the vertex of the cone. 


Consignor. One who sends mer- 
chandise (a consignment) to an agent 
(the consignee) to be sold. 


Corporation. A company author- 
ized by charter to transact business as 
a single individual. 


Coupons. Interest certificates at- 
tached to bonds. 


Days of Grace. Three days, in ad- 
dition to the time named in a note, al- 
lowed by law in most states for the 
payment of a note by its maker. 


Discount. An allowance deducted. 

True Discount. The difference be- 
tween the face and the present worth 
of a debt due at a future time without 
interest. The interest on the present 
worth. 


DEFINITIONS. 


Dividend. Profits of business di- 
vided among stockholders in propor- 
tion to their shares. 

Draft. An order sent by one party 
to another, requesting him to pay a 
specified sum to the order of some 
one named. 

Sight Draft. One payable when pre- 
sented to the drawee. 

Time Draft. One payable at a spec- 
ified time after sight or after date. 


Drawee. ‘The party ordered to pay 
a draft. 
Drawer. The maker of a draft. 


Duties or Customs. Taxes laid by 
the government on imported goods. 


Duty, ad valorem. A tax of a cer- 
tain per cent of the cost of imports in 
the country where they are bought. 


Duty, specific. <A fixed tax levied 
on imports according to weight, num- 
ber, or measure. 


Equation of Payments. The proc- 
ess of finding when several debts due 
at different times may be paid at one 
time without loss to either debtor or 
creditor. 

Exchange. A method of making 
payments or collections in distant 
places, by means of orders or drafts, 
without the actual sending of money. 

Extremes. The third and fourth 
terms of a ratio. 

Face of Note, Check, or Draft. 
The sum for which it is written. 

Gross Weight includes the mate- 
rial used in packing. 

Imports. Merchandise 
from a foreign country. 


brought 


235 


Indorsement. A signature on the 
back of negotiable paper. <A record 
of payment on the back of a note. 

Indorser. One who puts his signa- 
ture on the back of a note, check, 
draft, ete. 


Insurance. Compensation for loss 
by fire or other disaster. 

Annual Interest. Simple interest 
on the principal, and simple interest 
upon any overdue interest. 


Compound Interest. Interest reck- 
oned on both the “principal and the 
overdue interest added to the princi- 
pal as often as due. 


Exact Interest. Interest computed 
for parts of a year by taking the exact 
number of days and reckoning 365 to 
a year. 

Leakage and Breakage. A dis- 
count for liquors lost from casks or 
bottles during importation. 

Maker of a Note. 
makes the promise and signs it. 
promissor. 

Market Value. 
open market. 


The one who 
The 


Present value in 


Maturity. The time when a note, 
draft, or bond falls due and is legally 
payable. 


Means. The second and third 
terms of a ratio. 


Negotiable Paper. Notes, drafts, 
or other written obligations that may 
be bought and sold. 


Net Price or Cost. The price or 
cost after all discounts or charges have 
been deducted 


236 


Net Weight. 
packing material. 

Demand Note. One payable at the 
demand of the holder. ‘Time Note. 
One payable at a specified time. 

Interest-bearing Note. One con- 
taining the words ‘‘ with interest.’’ 

Promissory Note. A written prom- 
ise to pay a specified sum of money. 

Partial Payments. Payments in 
part of a note or debt. 

Par Value. Face value. 

Payee. The one to whom or to 
whose order a note, check, or draft is 
payable. 

Percentage. The process of com- 
puting by hundredths. The part of 
the base indicated by the rate per cent. 

Personal Estate. Property exclu- 
sive of land and buildings. 

Policy. The written agreement 
given to.the insured by the under- 
writers. 

Poll-tax. <A uniform tax on per- 
sons of a certain class. 

Port of Entry. A city or town 
containing a custom-house, where 
U. S. duties are paid. 


Weight exclusive of 


Premium. ‘Thesum paid for insur- 
ance. Excess of market value above 
par value. 


Present Worth. The sum that, 
at the present time, will pay a non- 
interest-bearing debt due in the future, 
without loss to either debtor or 
creditor. 

Proceeds or Avails of a Note. 
The sum for which the note is sold. 
Its maturity value less the bank dis- 
count. Net Proceeds. What is left 
after all charges have been deducted. 


DEFINITIONS. 


Proportion. An expression of the 
equality of two ratios. 

Pyramid. A solid, whose base is 
a regular polygon, and whose sides 
are triangles meeting in a common 
point, the vertex of the pyramid. 

Rate per cent. The number of hun- 
dredths used in finding a percentage. 

Real Estate. Land and buildings. 

Remittance. Money or negotiable 
paper sent to another. 

Share. One of the equal parts into 
which corporation capital is divided. 

Slant Height. The shortest dis- 
tance from the vertex of a cone or 
pyramid along the outside to the base. 

Sphere. A solid having a curved 
surface equally distant from the cen- 
tre at every point. 

Stocks. Shares in the capital of 
corporations. Government or corpora- 
tion bonds. 

Stock Certificate. A statement 
given by a corporation, showing the 
par value and the number of shares 
owned by a stockholder. 

Stockholders. Owners of the capi- 
tal or stock of corporations. 

Tare. An allowance for the weight 
of boxes, bags, etc., used in packing 
goods. 

Tariff. The list of dutiable articles 
with the rate assessed on each. 

Taxes. Money raised by govern- 
ment for public uses. 

Term of Discount. The time be- 
tween the day of discount and the day 
of maturity. 

Underwriters. 
panies. 


Insurance com- 


ORAL EXERCISES. yp 


' 
| 


MIscELLANEOUS EXERCISES. 


Oral. 


351.— 1. What makes a frac- 
tion larger ? 

2. What is it that shows how 
many times the denominator of 
a fraction is contained in its 
numerator ? 

3. In 2§ how many 40ths ? 

4. What is the ratio of 2 to 


5. Compare 164 with 31. 
6. Add 44, 12, and 23. 

7. From 7+ take 61. 

8. 16x $81=$2. 

9. 183% of $14.40 = $a. 
10. $17 is 2 of $a. 


352.— 1. 4 of 48 is 80% of x. 
3 is 2 of a. 

7:38 = 12:2. 

$6 is 3% of what ? 

DBO S20 a. 

0.006 x .0120 = a. 
ganar 

4.2 + 0.007 = a. 

0.42 + 70 = a. 

10. 0.042 + 0.0007 = a. 


7 ea Ser bo 


o 9 AD 


353.—1. 31% of 500 =a. 
2. $2 is what % of $800? 


38. $6is 3% of what number ? 

4. What per cent of a 2 ft. 
cube are 3 cubie feet ? 

5. A miller takes 4 quarts, 
or x% toll, out of every bushel. 

6. Spent $27, or 30% of my 
money. I had $a remaining. 

7. What rate of income have 
I on an $8000 house, rented at 
$ 40 a month. 


8. The tax is 15 mills on a 
dollar, or 2%. 
9. $160 is 0.8% of $a. 
10. Wages are increased 10%. 
Men that now receive $1.65 
formerly had $ a. 


354.— 1. $22.50 is what % of 
$ 750 ? 


2. 4+ of the selling price was 


gain. The gain per cent ? 
3. Selling at $3.50 I lost 
121%. What should I have 


gained by selling at $5? 

4. Sold stock at 120 that cost 
125. Loss % =z. 

5. 10¢ is 125% of what? 


6. Sold for $2, and gained 
400%. 


238 


7. A patent medicine costs 


16¢ a gallon and sells for $1 an | 


8 oz. bottle, or «% of the cost. 
(16 fluid ounces = 1 pt.) 

8. 4 yd. =< ft. y in. 

9. Cost of 2 lb. at $4 an oz. 

10. If from 4 
I had taken $2.50, 2 
money would have remained. 


355.—1. 41 x 22 x w= 25. 


2. 0.0375 = what common | 


fraction ? 
3. Change 2 of a week to 
hours. 


3. + mile = how many feet ? 
3 8 yes a 
pe eke =u Ty X 2 Se 
$ 25 x 3 


5. Divide 4 into three equal 
parts; 3 such parts make how 
much ? 


6. 28 of 24 = what? 
S) Pee eae ot pat OA bl eee 
1. (2s cs Gab ces 16 


8. 22 pounds = x ounces. 
9. 334 + 62 = what? 
10. Add 35, 33, 7%- 
1l. Repeat the table of length 
measures, giving the value of 
each unit in feet. 


356.—1. Sum of prime num- 
bers between 20 and 30. 
2. G.C. D. of 51 and 85. 


of my money | 
of my 


ORAL EXERCISES. 


3. L.C. M. of 6, 8, 16. 

4. 334% of 74 yards. 

5. 20% of half a million. 

6. 411 yd. at $ 0.124. 

7. The square root of 90,000. 
8. Surface of a 10-ft. cube. 

9. 20 da. interest of $120. 


10. Proceeds of a 60-day note 


: for $ 800, 9%, without grace. 


357.—1. Longest straight line 
that can be drawn on a6~x8 
sheet. 

2. Cost $18. Proceeds of 
sale $22. Gain per cent? 

3. Sold at 4% commission 
and earned $40. Sales ? 

4. $36,000 insurance at 3%. 
Premium ? 

5. Rate of taxation, $16 on 
$1000. Property tax, $112. 
Assessed value of property ? 

6. 8:24=74:2. 

7. Weight of an inch cube, 
10 oz. Of a 5-inch cube ? 

8. Cost of an acre at 10¢ per 
square foot. 

9. Square root of 27. 

10. Cost of 11 at $2 a dozen. 


of a 
The 


358.—1. Add 8 to 2 
number, and we have 32. 
number ? 


ORAL 


2. 62 yd. cost $53. 20 yd. ? 

38. Divide 50 in the ratio of 
7 to o. 

4. How many yards of carpet 
will be needed for a hall 9 ft. by 
15 ft. ? 

5. A building lot contains 
3,382 sq. ft. and is 100 ft. long. 
How wide is it? 

6. Cost of 4 gross at 8 for a 
dime. 

7. Divide 0.6 by 0.015. 

8. 4 x -V256 = 2’. 

9. 41s what per cent of 4? 

10. An are of 80° is what part 
of a circle ? 


359.— 1. A house bought for 
$3,200 was sold for $3,000. 
What per cent was lost ? 

2. $21 pays for how much 
insurance at 3% ? 

3. 121% of 64 is 334% of a. 

4. Find the interest of $400 
for 66 days at 6%. 

5. What is an agent’s com- 
mission at 14% on $3,000 ? 

6. What will pay a note of 
$ 500 that has been running 2 of 
a year at 9%? 

7. Cost of 5 yards at $1.50 a 
yard, 333% discount. 


EXERCISES. 


239 


8. Bought 3 for 4¢ and sold 
at the rate of 2 for 3¢. Did I 
gain or lose, and how much ? 

9. Bought for $10 and sold 
for $2.50. What per cent was 
lost ? 

10. What did I pay as an 
assessment of 2% on 15 $100- 
shares of stock ? 


360.—1. What is the perim- 
eter of an equilateral triangle 
whose side is 32 ft. ? 

2. How many feet in 43 rd. ? 

3. How many degrees does 
the long hand of a watch move 
over in 25 minutes ? 

4. What is my milk bill for 
3 weeks, 3 pints being taken 
daily at 7¢ a quart ? 

5. Cost of 49 lb. of flour at 
$ 5 per barrel. 

6. Cost of 123 yd. at $2 a 
yard. 

7. Contents of a piece of cloth 
containing 240 4-inch checks. 

8. What will 5 tons of coal 
cost at 4¢ a pound ? 

9. My weight increased from 
150 to 175. What was the per 
cent of increase ? 

10. The side of a rhombus 
measures 62 in. Its perimeter 
is what ? 


240 


361.—1. 9 months interest of 

$ 200 at 8%. 

2. How long will $500 be in 
oarning + of itself at 5% ? 

3. 8 yd.is what part of 7 yd.? 

4. What is the ratio 4 to 4? 

5. 134 = product of means ; 
one extreme is 27. The other ? 

6. The junior partner has 
the capital and $720 gain. Whe 
is the whole gain ? 

7. What time is it when the 
hours before noon equal the hours 
after noon ? 


& 
5 
t 


8. Eighty cents remains after 
losing 2 dimes and spending 4 of 
I had $~ at first. 

9. A can do =% of his work 
in 20 minutes. When will he 
finish if he begins at 10 A.m.? 


my money. 


10. How many yards of 3 in. 
bandages can be made from 7 
yards of yard-wide linen. 


362.— 1. If it is # of a mile 
to the post-office, how far will 
you travel in making 7 round 
trips to it? 

2. It is between 10 and 11, 
and the long hand has moved 
over 72° since the clock struck. 
What time is it ? 


3. How wide is a yard of rib- 
bon that contains 1 sq. ft. ? 


ORAL EXERCISES. 


4. To 2 add its cube. 

5. How many acres in a farm 
1 mile square ? 

6. From # take its square. 

Ce AN eee 

8. The inverse ratio of 2 to 3? 

9. The perimeter of a square 
10 A. field, or . of a sq. mile? 

10. What per cent of a day has 

gone at 2 P.M. ? 


363.—1. 0.075 x 0.4 = a. 
2. Z yd. at $2, and 12 yd. at 
371¢ cost $ a. 
3. What cost 42 yd. at $1.25 ? 
4. What shall I pay for 162 
lb. at 18%; and 872 lb. at 32¢? 
5. How many eggs ina basket 
if 12 are bad and 96% good ? 
6. How many 18 in. napkins 
will cover a table 2 yards square ? 
7. At $4 a dozen how many 
brooms can be bought for $ 2.50 ? 
8. $121 is 8L%.of what? 
9. 25 eggs at 25¢ a dozen ? 
10. What will $62.50 amount 
to in 200 months at 6% ? 


364.—1. Sold for $16 and 
gained 331%. What per cent 
should I have gained at $15? 

2. Gave $2 for a bunch of 
108 bananas. What per cent was 
gained by selling 2 for 5 cents ? 

3. 103 lb. cost $ 2.60; 21 lb. ? 


WRITTEN EXERCISES. 


4. Is it better to buy for $4 | 
and sell for $4.50, or to sell for | 


$8 W hat is bought for $7? 
5. A can roof a house in 3 
days; B, in 4 days; both in 2. 


365.—1. Add 12,4, 19°, 1OEs: 

2. From 108;% take 9614 

3. Find iy ve of 183 ee at 
$3.374. 16,5 doz. at $ 1663. 

4. 4650 Ib. coal at $5 AO; ar 
bbl. cement at 1.583; 4 off, eat 
discount. 

§. (6.25 +31) + (82 — 0.275), 

6. 4 of a lot of land sells for 
$ 1185.50. What is the rest of 
it worth at that rate ? 

7. 0.6825 less s4, = &. 

8. A can do + of a piece of 
work in a day. B can do the 
work in 5 days. Together they 
can do it in w days. 

9. From 3 mile take 1532 rd. 

10. From 1 cu. ft. take 19 
cu. in. 

ll. At $2 a yard, how many 
yards can be bought for $17 ? 

“ Find the cost of ¢ gal. at 

21¢ a quart; 25 eggs at 20¢ a 
reste 11 quires at $2.50 a 
ream ? 

13. How many square feet in 
ez of an acre ? 

14. 1728 sq. ft. = what frac- 
tion of an acre ? 


241 


366.—1. My profits of 35% 
were $210 one week, and $180 
the next week. My sales for 
both weeks were $ «a. 


2. Sold a house at an advance 


of 18% for $3240. What was 


its cost ? : 

3. Bought goods for $400 at 
a discount of 10 and 5%, and 
sold at the same price discounting 
5 and 10%. Did I gain or lose, 
and what ? 

4. Received 4% for collecting 
x dollars, and earned $ 22.50. 

5. Sold to A for $400; he 
sold to B for $500. His gain 
and mine were at the same rate. 
What was the cost to me? 


6. Bought 2000 lb. of sugar at 
48, and sold 18 lb. for $1. What 
was my gain per cent on a pound ? 
On the whole lot ? 

7. A dealer sold two bicycles 
for $100 each. On one he gained 
25%, and on the other he lost 
25%. Total gain or loss ? 

8. Had I better pay 
cash or $ 260 in 3 months ? 


9. Sold 60% of a lot of land 
for what 2 of the whole cost. My 
gain was v%. 

10. 
of 84? 


$ 256 


288 is %, of how many dths 


242 


367.—1. Describe one’s posi- 
tion after travelling 600° due 
west. 


2, 186,360? + 5,280 = 186,360 


3 16% x 66 _ 
124 


2 
at 22 Oe tO ayy 
47+ 45 + 130 — sis = &. 


5. The times at which suc- 


cessive miles of a bicycle race 
were completed were: 


h. min. _ sec. h. min. sec, 
Bo 10 Weed wane 0 cael 
PEE aS ORT ABS Le 
DVT. ts 207) an orm oer oe 


Find the average time for the 
five miles. 


6. If the wind through a tree 
pulls at each of 50,000 leaves 
with a force of 1 0z., what is the 
strain on the trunk ? 


7. 1,000 strips of canvas are 
wanted 58 in. long. In cutting 
every 20 strips 1 foot is wasted. 
How many yards must be bought ? 


8, Six dogs can lift 800 lb., 
but they can pull 4 sledge loads 
weighing together 9 times as 
much. Find by cancellation each 
dog’s share of each load. 


9, Capital $25,000, profits 
$3,680 or x%. 
10. $238 at 7% for 178 d. 


WRITTEN EXERCISES. 


368.— 1. Amount of a $525 
note, 1 yr.17 d. after date, at 5 %. 


2. Proceeds of a $2,750 note 
discounted 117 days before ma- 
turity, at 7%. 


3. Cost of putting a 6-inch 
coating of loam on a garden patch 
60 ft., by 36 ft., at 50¢ a cubic 
yard. 

4, My agent’s commission‘is 
21% on a purchase of hides, all 
expenses being covered by a 
remittance of $1132. Freight 
cost $25. Required his com- 
mission. 


5. Recovered $8,500 insur- 
ance on a total loss. What was 
my actual loss of 54% ? 


6. The surface of a hemi- 
spherical dome 15 ft. high is x 
sq. ft. 


7, A cubic foot of iron weighs 
450 Ib., a 15-in. cannon ball, a Ib. 


8, A cylindrical oil tank 34 
ft. by 22 ft. holds a gallons. 


9, The compound interest of 
$350 for 2 yr. 3 mo., at 4%, 
is $ a. 

10. A note for $500 running 
18 mo. has had two semiannual 
$100 payments. What is now 
due ? 


WRITTEN EXERCISES. 243 


369.—1. Find the average receipts for a full day at an office 
which is closed from Saturday noon to Monday morning, and takes 
in $ 72,147.84 in three weeks. 


2, What per cent of discount lets a $225 horse go for $1853 ? 


3. A conical sugar loaf 6 in. at the base and 1 ft. high would 
make a square cake 1 in. thick, x in. wide. 


4, What is the rate of duty on jewelry if $402.50 is charged on 
an invoice of $1,150 ? 


5, After my bill has been reduced by successive discounts of 
20% and 10%, I can pay it for $101.664. What was the gross 
amount charged ? 


6. A debt bearing 7% interest increases $ 61.52 by standing 2 yr. 
1 mo. How large is it at first ? 


7. In 1885 aluminum sold for $12 a pound. In 1895 it can be 
bought for 35 cents. What is the per cent of reduction? 


8, China is to pay Japan a war indemnity of $160,000,000. 
One half is to be paid at once, and the remainder in 6 years, with 
simple interest at 6%. What will be the amount of the second 
payment ? 


9, The successful candidate received 6,750 votes. His opponent 
had 5,825. How were every 100 votes divided ? 


10. What is the tax rate on $1 where $73,000 are to be raised on 
a valuation of $5,309,090. 


11. If the Great Pyramid covers 13 A., how long is one side? 
How much more stone would be required to make a prism of the 
same _base and altitude ? 


12. How large a square could be made from the cover of a 3-inch 
ball ? 


" 13, $1,200 is due with interest and without grace in 1 year. If 
half is prepaid now, how much will discharge the debt at the end of 
6 months ? 


244. WRITTEN EXERCISES. 


370.—1. A man, engaged at $45.00 a month, worked from Dec. 
14 to May 9. His wages were « dollars. 

2. I bought 1185 lb. of hay, at $19.50 per ton, for a dollars. 

3. The assessed value of.an estate is $55,000. The house is 
assessed for $28,200. The property has a frontage of 36 ft. and - 
is 126 ft. deep. The land is assessed for @ dollars per foot. 

4. Water freezes at 32° above 0 (Fahrenheit) and boils at 212°; 
but, according to the Centigrade thermometer, it freezes at 0° and 
boils at 100°. 68°.F. equals 2° C. 

5. Bought an estate comprising 19,367 sq. ft. for $27,800. I sold 
the buildings for $2685, and paid the interest on a mortgage of 
$ 12,500 for 3 years at 5%, also $565.00 for taxes. I then sold the 
land at $1.45 per sq. ft. I gained or lost x dollars, not counting 
interest on my money. 

6. The last reading of my gas-meter was 67,300 cu. ft. The pre- 
vious reading was 64,900. At $1.50 per thousand, with a discount 
of 15 ¢ per thousand ft., my gas bill was @ dollars. 

7. By working every week day in the month of August, which 
began on Sunday, a boy picked 24 bu. of berries, which he sold for 
61¢ a quart. He earned & cents a day. 

8. Mr. Brown is taxed for a dollars. His tax bill is $110.00, 
including $ 2.00 for poll tax, the rate being 9 mills on a dollar. 

9. A man received a dollars for sawing a pile of cord wood, 56 ft. 
long and 4 ft. high, @ $1.50 per cord. 

10. Two coaches set out on the same journey. The red one com- 
pletes 2 of it, the yellow one, 4. Which is greater, the distance 
between them or the distance the yellow one has to go ? 

11. Two ships sail the same course in opposite directions. One 
has finished 53, the other $ of it. What part of the whole distance 
hes between them ? 

12. For 1895 a clerk’s salary was $1242. It had been increased 
10 % in 18938 and 8% in 1894. What was it in 1892? 

13. What was the per cent of increase from 1892 to 1895 ? 


WRITTEN EXERCISES. 245 


371.—1. How far is a man from his starting-point, who travels 
west 48 miles, then due north 62 miles, and then east 14 miles ? 

2. Show that 90 days’ interest on any sum of money at 4% is 
found by removing the point two places to the left. 

3. 15° difference in longitude makes an hour’s difference in the 
time of sunrise. The sun rises at 4.50 at A; at B, 80° farther 
west, it is then what o’clock by sun time ? 

4. Each side of a triangle measures 30 ft. Its area is 2. 

5. A lot containing 30,000 square feet is } as wide as it is long. 
What will it cost to fence it at 121¢ a running foot? (What part 
of the lot makes a square ?) 

6. A cubic foot of distilled water weighs 1000 oz. The weight of 
any substance, as compared with that of an equal bulk of pure 
water, is called its specific gravity. The specific gravity of water 
is 1; of gold, 19.258; of cork, 0.240; of pure alcohol, 0.794. What 
is the specific gravity of granite that weighs 1652 lb. to the cubic 
foot? 7. What does a gallon of alcohol weigh ? 

8. Bought railroad stock at 1148, and sold at 1174, the purchaser 
in each case paying 4% brokerage. What was my profit on 200 
shares ? 

9. # carpeting is used for a room 20 feet square. The waste in 
matching is 6 inches to a strip. The cost at $1.75 per yard is $a. 


$ 600 — Chicago, Mareh /9, 1895. 

Four uontha after date I promise to pay to the order 
A eee Pee act hs POMC CACO B6 e 
1 aie NOC 6ight /Fundred _____.......----.----- Dollars 
at the Aoth Bank. 


Value received. 


Uetor Robinacn. 


10. This note was discounted April 17, at 5%. Proceeds? 


246 WRITTEN EXERCISES. 


372.— 1. If 2 of a day’s wages are $1.40, what shall be paid for 

7, of a day ? 

2. s5 18 $% of what number ? 

3. A merchant gains 74% by selling goods for $2128.50. What 
did they cost ? 

4. 24 carat gold is pure gold. What per cent of alloy in 14 
carat gold? 

5. Bought 300 gross of files listed at $14.25, at 20, 10, and 5 off. 
Required the cost. 

6. A dealer obtained $480 for a piano on the list price of which 
he had discounted 50%. He still made a profit of 20%. What did 
it cost him ? 

7. Bought $15,000 worth of goods on 4 months, and sold imme- 
diately for $14,900 cash. Money being worth 5%, what did I gain ? 

8. The diameter of a circle is 20 in. Find the side of an 
approximately equal square. 

9. How many cubic inches will a box contain, 9 by 12 by 15 in., 
outside measure, if it is made of $ in. stuff. 

10. A was to receive 3, B 2, and C the remainder of a bequest of 
$18,000. C died, and his share was equitably divided between A 
and B. What did each receive ? 

11. What sum will cancel a 5% note for $763, dated April 19, 
1894, and maturing Aug. 11, 1896? 

12. If a speculator saves himself from failure by borrowing 
$50,000 for 4 hours at 1% per day, how much interest does he 
pay ? 

18. I bought cloth by the meter (39.37 in.), and sold at the same 
price per yard. What per cent did I gain ? 

14. The 600 shade trees in the common are increased in five 
years to 1350. What is the yearly per cent of increase ? 

15. A 17 in. square of basket work is plaited into 289 small 
squares. How wide is the material used ? 


WRITTEN EXERCISES. 247 


373.—1. What is the semiannual dividend on 15 $50 shares of 
7% vailroad stock ? 

2. 4% government bonds yield an annual income of $1000. 
What is their face ? 


3. How many shares of stock that sell at a premium of 57} can 
be bought for $5,000, no brokerage ? 
4. $120 yields $8 annually. What rate per cent is this? 


5. What is the market value of 25 shares of New York Central 
stock at 463 premium ? 


6. Paid $200 for a share of telephone stock, and received an 
annual dividend of $10. This was x per cent on my investment. 

7. What does a $6,500 mortgage yield semiannually at 54% ? 

8. Stone & Co. purchase $1,500,000 city bonds at 22% premium, 
one third of which they sell at a premium of 4?, one third at 53, 
and the remainder at 4,%. What is their profit, the cost of selling 
being 3 % ? 

9. Solda7% mortgage for $3,000 at 25% premium, and bought 
6% railroad stock with the proceeds. Did I increase or lessen my 
income, par of stock being $50? 

10. Which is the better investment, 5% stock at 120, or 6% at 

Tyee ery) ee leg 

11. $528 is my annual income from 4% stocks. What is it 
worth at $125 per share ? 

12. A charitable organization has the following assets. What is 
its annual income? $6,000 in 5% mortgages; $25,000 in govern- 
ment 4’s; 900 shares 4% stock; 2,000 $10 shares yielding 34% 
semiannually. 

13. A watch appears to be 5 h. slow because of a change of longi- 
tude. Has it been taken east or west, and how many degrees ? 


14. A side of a hexagon is 6 in. Find the distance between its 
parallel sides. 15. Give the area of the hexagon, and that of the 
largest circle to be drawn within it. 


248 WRITTEN EXERCISES. 
374. — Questions 1. Is a fraction a number? 2. Mention 
on Principles and four integral units of different kinds and four 
Methods. fractional units of different values. 


3. Show how the fundamental process of 
arithmetic may be reduced to two, — putting numbers together and 
separating a number into parts. 

4. Show the effect of moving ‘the decimal point. 

5. What is the difference between a decimal and a common frac- 
tion ? | 

6. Show why doubling the denominator halves a fraction. 

7. Explain the principle on which the change of a fraction to 
larger units 1s based. 

8. How is the number of decimal places in the product deter- 
mined? Explain why this is so. 

9. When does a note begin to draw interest? 10. Why are 
notes sometimes indorsed ? 

11. Why does multiplying the antecedent multiply the ratio ? 

12. What is the difference between specific and ad valorem duties ? 

13. Show why pointing off two places from the right of a prin- 
cipal gives 2 months’ interest at 6%. 

14. Why do leap years contain 366 days? 15. What are the natural 
divisions of time? 16. Why will your watch be an hour fast if you 
go 15° west ? 

17. Why should a 5-inch ball be 125 times as heavy as a 1-inch 
ball of the same material ? 

18. Why should a pound of feathers be heavier than a pound of 
gold ? 

19. How would you show the ratio of circumference to diameter 
to be 3.1416 ? 

20. What is meant by saying that the specific gravity of cork is 
0.240 ? 

21. Show by a drawing that the “square of the hypotenuse equals 
the sum of the squares of the two short sides of a right triangle.” 


FROM EXAMINATION PAPERS. 249 


375.—1. If 38} bushels of turnips cost $283, what will 124 
bushels cost ? 


2. The cost of 50 gallons of molasses is $25. If + is lost by 
leakage, and 20 gallons are sold at 621 ¢ a gallon, at what price per 
gallon shall the remainder be sold to gain $5? 


3. A gentleman paid 3 of his money for a farm; had he paid 
$100 more, he would have paid 3 his money. The farm cost $ 2. 


4. An agent received $52 as his commission at 4% for buying 
flour at $5 a barrel. How many barrels were bought ? 


5. At 24% discount, what shall be paid for 4% stock so that the 
annual income shall be $ 2400 ? 


6. A house was sold for $ 1850 at an advance of 15% on the cost. 
What would it have brought at a gain of 20% ? 


7. Three drovers rent a 9-acre field at $5 an acre. A puts in 6 
cows for 2 months; B, 9 cows for 1 month; and C 12 cows for 2 
months. What should each pay ? 


8. 75% of a farm is arable; of the remainder, 80% is pasture, 
and the remaining 3 A. 20 sq. rd. is waste. What is the area of the 
farm ? 


9. How long must a stick of timber be which is 14 feet square 
throughout its length and whose volume is 100 cubic feet ? 


10. A rectangular field is 64.8 rods long and 56.05 rods wide, and 
a square field is of equal area. At $1.10 a rod, 
how much more will it cost to fence one than the 
other ? 


Cc 


11. The accompanying figure represents a sur- 
face containing 15,376 square feet. Find the length 
of mn, connecting the middle point of two sides. 


12. Memorandum. — Note for $1824, dated 
Chicago, Oct. 10, 1895. Payment of $500 was 
made April 25, 1896. What was due Jan. 2, 1897, interest at 
6% ? 


nm 


a n b 


250 PROBLEMS. 


376.—1. From a field containing 50 A., I sold a corner 100 rods 
long and 40 rods wide. What % remained ? 

2. I paid $ 25 for linoleum, at $1.25 per square yard. The length 
of my floor was 15 ft. What was its width ? ~ 

3. What is the rate of interest when $62.50 is paid for the use 
of $800 for 1 yr. 5 mo. ? 

4. I paid 3% brokerage on a sale of wheat at $1.15 a bushel. 
The brokerage was $63.48. How much wheat was sold ? 

5. Sold my carriage at 30% gain, and with the money bought 
another, which I sold for $182 and lost 123%. What did each 
carriage cost? 

6. A owns $ of a manufacturing plant. The plant is valued at 
$48,870. A sells a part of his interest for $10,860. What part 
of his interest does he sell ? 

7. What rate of interest do I get for my money when I buy 6% 
bonds at 108 ? 

8. A horse tied to a stake can graze over 2181 sq. yd. of surface. 
How far from the stake ? 

9. How many acres in a street 14 miles long and 4 rods wide ? 

10. A man with $27,000 has his choice of two equally safe 
investments, —a 5% mortgage, or bonds at 185 paying 7% on the 
par value. Which would you advise him to take ? 

11. An irregular piece of land, containing 4 A. 621 sq. rd., is 
exchanged for a square piece at the same price per foot. What is 
the length of a side of the latter ? 

12. Find the lateral area of an octagonal pyramid whose slant 
height is 30 in., each edge at base 20 in. 

13. How long is a chain of 1600 one-inch links made from ;,-inch 
wire ? 

14. A coal-bed 10 ft. thick covers a sq. mile. How many 15 T. 
car-loads, allowing 45 cu. ft. to a ton? 


FROM EXAMINATION PAPERS. 25) 


377.—1. The product of two numbers multiphed by 4 is 0.0005. 
One of the numbers is 0.05. What is the other ? 


2. What will it cost to cement a cellar bottom 36 ft. long, 28 ft. 
7 in. wide, at 96 a square yard ? 

3. A dry article was exposed to dampness, and absorbed 4 ounces 
of water. It then weighed 9 pounds. What per cent of this weight 
is water ? 

4. Find the value of the liquid in a cylindrical can 11 in. in 
diameter and 21 in. high, at 20 cents a pint. 

5. A piece of cloth, when measured with a yard-stick that is 2 of 
an inch short, appears to be 18 yards long. What is its true length ? 

6. A square field, containing 274 A., has a diagonal path across 
it # rods in length. 

7. Sold 2 of an article for what 2 of it cost. What was the gain 
per cent ? 

8. Find the surface of a sphere 25 inches in diameter. 


9. A square court, whose side is 42 yards, 1s paved with 28,224 
square tiles. Find the dimensions of each tile. 


10. A merchant buys goods for $1125. He sold 4 at an advance 
of 25% on the cost, 2 at an advance of 125%, and the remainder at 
one-half their cost. What was his profit ? 

11. What will $1350 amount to in 125 days at 44% ? 


12. Sold a horse so that + of the gain equalled ;?, of the cost. 
What was the gain per cent ? 


13. A, B, and C trade in partnership. A furnished 4 the capital, 
and is to have 4 the gain for extra services. B and C furnished 
$3000 each, and the gain is $5760. How is it divided ? 


14. 1.92 + 0.0048 — 0.0048 + 192 = what ? 


15. Twelve persons hired a boat fora certain sum. Four of them 
withdrew without paying, and thus the expense of each of the 
others was increased by $2. What was the rent of the boat ? 


252 PROBLEMS. 


378.—1. A merchant bought 3 yards for $2 and sold 2 yards for 

$3. What was his gain per cent? 

2. What is the entire surface, in square feet, of a box 18 in. by 
16 in. by 14 in. ? 

3. A coat cost $8. How shall it be marked to lower the price 
20% and still gain 20% on the cost ? 

4. How much will it cost for steel rails to lay one mile of double- 
track railway at the rate of $ 56.25 for 100 feet of rail ? 


5. Paid an attorney $ 18.16 for collecting $72.64. How much 
would he need to collect to earn $ 5,000 a year at the same rate ? 


6. A tree 100 ft. high casts a shadow, on level ground, 75 ft. 
long. How far from the-end of the shadow to the top of the tree ? 


7. What is the cost of 35 miles of telephone wire at 40 cents a 
pound, supposing a pound to stretch 20 feet ? 


8. If 25 men can do a piece of work in 15 days, how many men 
will be needed to do 4 times as much in ¢ the time ? 


9. At8 cents a foot, what will be the cost of a board 12 feet long, 
10 inches wide at one end, and tapering to a point ? 


10. I sold two cows at $45 apiece. On one I gained 20%, and on 
the other I lost $17.50. For what should I have sold the two to 
gain 54% ? 

11. How many square feet are there in a tight board fence, 4 feet 
6 inches high, round a circular lot 10 rods in diameter ? 


12. Leaving ? of my money at home, I spend 5% of the rest for 
egos that cost me 29 cents per dozen. I bought eggs enough to fill 
8 baskets, 5 dozen to a basket. How much money had I at first 2 


18. Into a gallon of water I put a pint of alcohol. I then drew 
off a quart of the mixture. What per cent of the alcohol did I 
draw off ? 


14. Assuming a meridian to be 12,500 miles in length, how many 
miles is it from one tropic to the other ? 


FROM EXAMINATION PAPERS. ; 953 


379.—1. The list price of a carriage is $260. I am allowed 20% 
and 10% discounts. What is the net price ? 
2. The net price of a mowing machine is $ 158.40, and the trade 
discounts are 20% and 10%. Find the list price. 


3. If by selling an article for $9.50 I lose 5%, for how much 
should I sell it to gain 5% ? 


4. A building was insured on # its value at #%, the premium 
being $19.50. What was the owner’s loss when it burned ? 


5. Purchased stock at 80, and realized 124% from my investment 
by getting one annual dividend and selling immediately at cost. 
What was the stock paying ? 


6. A bookseller buys books from the publishers at 40% off 
the list price. He sells a set of Thackeray’s works which list at 
$ 30 at a discount of 20%. What per cent does he make ? 


7. A man, who had been paying $35 a month rent, borrowed 
$ 5,000 at 5% and bought the house. Instead of rent he now pays 
the interest on his borrowed money, $60 a year taxes, $12 water 
rate, and $50 for repairs. Find his gain or loss per year. 


8. A bought a ship for $8,750; expended upon her for repairs 
the amount of 10% of her first cost; paid 24% for insuring her on 
2 of her total cost; and she burned at sea. How much did he pay 
on her account altogether? How much did he receive ? 


9. A farmer has a trapezoidal lot whose parallel sides are 35 rd. 
and 84 rd. The perpendicular distance between the parallel sides 


LP 


is 832 rd. How much will it cost to plow the field at $2.50 an acre ? 


10. The owner of a wrecked steamboat sells her wrought iron 
crank shaft, which is 18 ft. long and averages 8 in. in diameter, 
(a, 7 cents per lb. How much does he realize if there are 483.4 Ib. 
in a cu. ft. ? 


11. I have $1,000, with which I buy as many watches as possible 
at $43.75 each. How much more money do I need to buy one 
watch more ? 


954 | PROBLEMS, 


380.— 1. If 32 men can lay city railroad tracks a distance of 600 
rods in 15 days, in what time can 40 men lay 840 rods of tracks ? 

2. A house rents for $40 a month, the annual expenses on it are, 
—taxes $92.50, water rate $20, and repairs $60. The landlord 
has five per cent clear profit. What did he pay for the house ? 

8. On goods invoiced at $8,000 the duty was 15%; on others 
invoiced at $17,000 the duty was 25%; the total cost of another 
invoice, including the duty of 30%, was $12,090; what was the 
whole amount of duties ? 

4. After buying some goods, a merchant lost 20% of them by 
fire. He sold the remainder at again of 331%, receiving $250.75 
more than he paid for the whole. What did the goods cost ? 

5. The perimeter of one square field is 500 feet, and of another 
360 feet; what would be the perimeter of a square field that is 
equal in area to both ? 

6. A cylindrical tub is 34 feet in diameter; the recent heavy 
rain filled it with water to the depth of six inches, making how 
many gallons, 231 cu. in. to a gallon? 

7. How many board feet in a stick of timber 27 ft. long, 11 in. 
wide at one end, 8 in. wide at the other, and 12 in. thick ? 

8. I sold a flouring mill, receiving 45 per cent of the price in 
cash, and invested 75 per cent of the sum received in a city lot worth 
$ 2,160. For how much did I sell the mill ? 

9. In three successive years a manufacturer invests in his busi- 
ness sums equal to 15 per cent, 10 per cent, and 20 per cent, respec- 
tively of his capital at the time of the investment, and in the fourth 
year he loses 50 per cent of his capital. At the end of the fourth 
year his capital is $23,695.98. How much was his original capital ? 

10. I ordered my agent to buy flour which I afterward sold at 20 
per cent profit and gained $1.56 per barrel. If my agent’s rate of 
commission was 4 per cent and his total commission $ 23.40, how 
many barrels did he buy ? 

11. 4,000 copies of a book, containing 420 pages, were printed from 
650 reams of paper; how many reams of paper would have been 
required to print 7,000 copies, containing 528 pages of the same size ? 


FROM EXAMINATION PAPERS. 955 


381.—1. If aman can dig +4 of a ditch in 22 days, how long will 
he be in finishing it ? 


2. At $3} a cord, a pile of 4-ft. wood 82 ft. long cost $174. 
How high was the pile? 


3. At $60 an acre, what is the value of land whose two parallel 
sides are 25 rds. and 75 rd., the distance between them being 55 rd. ? 


4. Every person breathes on an average 28 cu. ft. of air an hour. 
How many hours will the air in a room 16 ft. by 12 ft. by 94 ft. 
last 12 men? 

5. A note for $550, dated Jan. 1, 1896, due in 4 months, was 
discounted Jan. 17, at 7%. Proceeds ? 

e 


6. Find the square root of (454, — 4.084 + 8) + 5000. 


7. Bought 150 shares of stock at 2% premium, and sold at 14% 
discount. Brokerage in each case 4. Loss ? 


8. How many square inches are left of a sheet of paper 14 in by 
21 in. after the largest possible circle is cut out of it? 


9. A grocer bought 75 lb. of soap at 64 cents a pound. While on 
hand it dried away one-fourth in weight. He sold it at 8} cents a 
pound. What was his gain or loss per cent ? 


10. A broker bought 60 shares of stock, par value $50, at 94, and 
after receiving a dividend of 3% sold them at 1003. What did he 
gain ? 

11. Foster & Rich began business Jan. 1, 1894, with a capital of 
$ 7,500, of which Rich furnished $5,000. Foster was to have $ 800 
a year salary from the profits. How did they divide $2,000 of 
profit Dec. 31, 1894 ? 

12. The wheels of a bicycle are 30 in. in diameter; the gearing 
is such that each wheel makes two revolutions to every turn of the 
pedals. . How many times will each pedal turn in a ride of one mile ? 

13. The end of a square prism 25 ft. long contains 625 sq. in. 
What is the area of one side? 


256 PROBLEMS. 


382.—1. The 15 minutes’ recess is 5% of the time devoted to 
study and ‘recitation in a certain school. How many hours are 
so employed ? 


2. Mr. James owned 2 of a mill and sold.18% of his share for 
“$1,296, which was 121% more than it cost him. What was the 
value of the mill at this rate ? . 


3. A man bought a pair of horses for $400, which was 20% less 
than their real value, and sold them for 25% above their real value. 
What was the selling price ? 

4. A merchant having a debt due him of $6,424, compromises 
for 80%. What will he receive if he pays his agent 21% for col- 
lecting ? 


5. A has in bank $8,000, and loans 55 of it to B for 2 years 3 
months 6 days, at 9%. How much does B owe him at the expiration 


of the time? 


6. I bought a house for $ 2,500, and sold it so that 20% of the 
selling price was profit. What did I receive for it? 


7. $800. CLEVELAND, June 1, 1894. 

Six months after date I promise to pay to James Hutton, or 
order, Eight Hundred Dollars, for value received, interest 6%. 
Discounted Sept. 15, at 4%. Find proceeds and bank discount. 


8. When a post 11.5 ft. high casts a shadow 17.4 ft. long, a 
neighboring steeple casts a shadow 63.7 yards long. How high is 
the steeple, supposing the ground to be level ? 


9. What will it cost to dig a well 5 ft. in diameter and 30 ft. 
deep, at 55 cents a cubic yard, estimating to the nearest cubic yard ? 
‘10. A box 6 ft. long, 4 ft. wide, and 3 ft. deep, is full of oats. 


What is the value of the oats at 30 cents a bushel ? 


11. A man, owning ~ of a farm, sold 12 per cent of his share to 
B, and then the remainder to C for $20,020. What was the value 
of the whole farm ? 


FROM EXAMINATION PAPERS. Q57 


od 


383.—1. If 44 tons of coal cost $302, how many tons may be 
bought for $100? 

2. A, B, and C bought a horse for $100, and sold it for $150, 
by which A gained $18, and B $19. How much had A, B, and C 
each paid toward the purchase ? 

3. What ought I to pay a broker for two $400 6’s at 1064, three 
$100 44’s at 1043, and five $500 4’s at 994, with his brokerage of 
+% in addition to the prices named ? 


4. A dealer bought 100 bushels of potatoes at 40 cents a bushel. 
If he lost 50% of them, at what price per bushel must he sell the 
remainder to gain 20% on his investment ? 

5. In what time will the interest of $884.60 at 5% equal one- 
twelfth of the principal ? 


6. A man bought cloth costing $2454. The number of yards 
and the number of dollars per yard were the same. Find the number. 


7. Find the cost of 40 planks at $50 per M, board measure, each 
plank 12 ft. long, 2 in. thick, 20 in. wide at one end, 16 in. at the 
other, and tapering regularly. 

8. How many gallons of water, each 231 cubic in., in a circular 
cistern, 4 ft. 4 in. in diameter, the water 16 feet deep ? 


9. Find the result of 1.76 x 49.647 ~ 0.0088. 


10. A bin is 10 ft. square at the bottom. How deep must it be to 
contain 1,000 bushels of grain, allowing 1.244 cubic ft. to a bushel? 


ll. Find the-ratio of lighting surface to floor surface in a room 30 
by 35 ft., with 4 windows, each 3 ft. by 8 ft. 9 in. 


12. Find the cost of boards to make two contiguous rectangular 
rooms, each of whose dimensions is 10 ft., at $18 per M, allowing 
for a single partition between the rooms, no floors, and 25% waste 
in the lumber purchased. ' 


13. Each side of a pentagon measures 5 ft.; the perpendicular 
distance from the centre to one of the sides is 43 ft. What is the 
area of the pentagon ? 


258 PROBLEMS. 


384.— 1. B endowed a professorship with a salary of $ 2,000 per 
annum. What sum must he invest at 6% to provide this salary ? 

2. A man has $8,000 which he wishes to loan for $ 600 per year 
for his support. At what per cent must he loan it? 

3. On a note for $425, at 8%, dated March 25, 1898, are the 
following indorsements: June 1, 1898, $75; Dec. 30, 1898, $120. 
What is due Sept. 1, 1899 ? 

4. A railway train runs 2 of a mile in 4 of a minute. Find its 
rate per hour. 


5. A pupil who attended school 68 days during a term was marked 
85% for attendance. How many days was he absent ? 


6. The spire of a church is an octagonal pyramid. Suppose it to 
be 80 ft. high, each side 6 ft. at the base. Find the cost of covering 
it at $6 a square. 

7. A’s farm is 240 rods wide; he sells 18 acres off one end. How 
much shorter is his farm than it was before ? 


8: A wagon weighed 638 lb.; when loaded with wheat it weighed 
4,313 lb.; the box was 9 ft. 4 in. long, 3 ft. 6in. wide. How deep 
was it? 

9. If a cubic foot of iron weighs 500 lb., what will a cannon ball 
6 in. in diameter weigh ? 

10. What will it cost to gild the cannon ball at 3¢ a square inch. 


11. Which will be the greater, 6% simple interest on $3,000 for 
3 yr. 6 mo., or compound interest at 5% on the same sum for the 
same time ? 

12. A certain house was built by 40 workmen in 48 days, but, 
being burned, it is required to rebuild it in 50 days. How many 
men must be employed ? 


13. A garrison of 600 men has bread enough to allow 16 ounces 
a day to each man for 15 days; but, the garrison being reinforced by 
200 men, how many ounces a day may each man have in order that 
the bread may last 20 days ? 


FROM EXAMINATION PAPERS. 259 

385.—1. A miller keeps + of the wheat for grinding it. How 

many bushels, pecks, and quarts must a man take to mill in order 
to carry back the flour from exactly 13 bushels ? 


2. At what price must I mark a set of furniture costing $ 31.20, 
so that I may take 4% from the marked price, and still make 16% ? 


3. A grain merchant buys through his agent 4,500 bushels of 
wheat at 621¢ per bushel, commission 2%. He insures it for $ 2,800 
at1i%. He had borrowed the money to buy the wheat Mar. 1, 1892. 
He receives his insurance money Mar. 1, 1895, and pays his note, 
which has drawn interest at 6%. Find his total loss. 

4. A man bought Pacific R. R. bonds at 107, sufficient to give an 
annual income of $252 at 6%. What did he pay for them, broker- 
age 1% ? 

5. Write a note in which Benj. Simpson agrees to pay Sam’! Hill, 
or order, $ 240, with interest at 6%. Date Sept. 1, 1895, to run a 
year. What will Mr. Hill get at the bank for the note May 10, 1896 ? 

6. How many cubic feet of iron in a gas pipe 20 ft. long, 18 in. 
in circumference, with 3 in. bore ? 

7. What per cent should I receive on my investment if I should 
buy, at 10% discount, stock which pays an annual dividend of 44% ? 

8. Find the cost of papering the walls of a hall 36 ft. long, 24 ft. 
wide, and 18 ft. high, with paper 14 ft. wide at $2.50 a roll of 16 yd., 
allowing 64 sq. yd. for doors and windows. 

9. Three men, A, B, and C, engage in partnership; A puts in 
$1200, B $1500, and C $1900. They gain $350. What is the 
share of each in the profits ? 

10. The owner of 53 of a mine sold 5%, of his share for $40,500. 
What should he who owns 2 of the mine get for 3 of his share ? 


11. What is the amount of $897.25, at 6% simple interest, from 
Sept. 19, 1891, to March 13, 1894? 

12. A bookseller buys a book whose catalogue price is $3.50, at a 
discount of 20% and 5%, and sells it at 10% above the catalogue 
price. . What per cent profit does he make ? 


260 PROBLEMS. 


386.—1. An estate sold for $45,000, which was 374% below the 
appraised value. What was the appraised value? 

2. When chairs are sold at $4.80 per dozen, with a discount of 
5% for cash, what is the cash value of 200 chairs ? 

3. A owned 2 of a factory, and sold 3 of his share to B, who sold 
1 of his share to C, who sold 3 of what he bought to D. What part 
of the factory did each then own ? 

4. The gross amount of a bill of goods is $750.35, and the rates 
of discount are 10%, 10%, and 5%. What is the net cash to the 
purchaser ? 

5. When the duty on a quantity of lace at 30% ad valorem was 
$115.80, what was the cost of the lace, and the duty in frances at 
$ 0.193 per franc ? 

6. A man has settled on his wife $1,200 income a year. What 
sum must he invest in government 4 per cent bonds at 107} to pro- 
duce the required amount of income ? 

7. A yard is 84 feet long and 80 feet wide. What is the length 
of a clothes line that will reach from one corner to the corner diago- 
nally opposite ? 

8. A note of $1,050, dated Feb. 13, 1895, due six months after 
date, and drawing interest at 6 per cent, was discounted at a bank 
at 8 per cent, May 13, 1895. Find the proceeds. 

9. I bought two houses for $11,700, paying 25 per cent more for 
one than for the other. I sold the cheaper house at a profit of 20 
per cent, and the higher priced house at a profit of 162 per cent. 
What.was my total gain ? ‘ 

10. A and B offer the same quality of goods at the same lst price. 
A offers a discount of 15 per cent and 5 per cent, and B offers a 
single discount of 20 per cent. Of whom will it be the more advan- 
tageous to buy, and how much would be saved on a bill, the lst price 
being $ 185.50 ? 

11. The ice on a circular. pond is two feet thick. If the pond is 
1,000 feet in circumference, how many cubic feet of ice does it con- 
tain ? 


FROM EXAMINATION PAPERS. 261 


387. — From Civil 1. If $ of a pound of butter cost 338, of a 
Service Tests. dollar, eh will 32 Ib. cost ? 
2. If 2 of a ton of coal cost $ 2.56, how 
much will 14 tons cost, the latter being 25% cheaper per ton than 
the former ? 


3. A house rents for $30 a month, and the owner pays $75 a 
year for taxes and repairs. What is the value of the house if his 
net profit is 5% per annum ? 


4. What number must be added to the sum of 32, 7, 3%, to make 
81? 
5. An army officer, in preparing for a march of 6 weeks, buys 
oats for 32 horses at 572 cents a bushel. Hach horse will eat 2 of a 
bushel a day. How many bushels does he buy and what is the total 
cost ? 


6. Change 1% to the form of a decimal and multiply it by .035. 


7. The steamer “City of Paris” made the run from Queenstown 
to Sandy Hook, 2,788 miles, in 5 da. 19 hr. and 18 min. What 
was the average rate of speed per hour ? 


8. How many tons of coal can be put into a bin 12 ft. square 
and 6 ft. high, allowing 55 |b. of coal to a cubic foot, and 2,240 Ib. 
to the ton ? 


9. What is the cost of 8 pieces of paper, each 151 yards, at 
‘$1.75 per piece of 11 yards? 
10. If a merchant’s gain on $15,000 worth of sales is $ 937.50, 
what amount must he sell to gain $ 5,060 ? 


11. The cost of insuring a warehouse at 14% is $72 a year, and 
the cost of insuring its contents at 21% is $129.42. What is the 
whole amount insured ? 


12. In 1888 a railway company paid dividends on its stock as 
follows: 3 mo. at the rate of 7% a year, 6 mo. at the rate of 6% a 
year, 3 mo. at the rate of 2% a year. What did the dividends 
amount to that year on 28 shares of stock ? 


262 PROBLEMS. 


388.— 1. Three gross of lead pencils are divided equally among 
the clerks in a post-office, giving to each clerk 11 and leaving a 
remainder of 14 pencils. How many clerks are there in the office? 

2. A physician whose charges are $2 a visit, made on an 
average 5 visits per day in a year of 365 days. He collected 55% 
of his charges and saved $2 out of every $5 collected. At this 
rate how much did he save in 2 yr. and 6 mo. ? 

3. In an office employing 95 carriers, each carrier loses 20 min- 
utes a day in idle talk. Suppose the average salary of each to be 

2.50 for 10 hours’ work, what is the cost to the government of the 
lost time each day, and what will it amount to in a year of 313 
working days ? 

4. A merchant imported 120 tons of English iron, costing 14 
pence per pound, on which he paid a duty of 20%. The freight was 
5 shillings sterling per ton. What was the total cost in United 
States currency? (The ton equals 2,240 lbs. The pound sterling 
equals $ 4.8665.) 

5. Owing to a deficiency in the appropriation bill, the salaries 
of the clerks in a bureau were reduced 18% for the last quarter of 
the fiscal year. How much did a clerk who was paid $287 for the 
last quarter receive during the whole fiscal year ? 

6. A grocer sells goods to a customer for $552 by weights 
averaging 151 ounces to the pound, and afterwards sells goods for 
$ 320 by weights averaging 16} ounces to the pound. How much 
does the grocer make or lose by the false weights ? 

7. A merchant buys 42 gallons of whiskey at $2.50 per gallon, 
and keeps it for 5 years. He then finds that he has lost 7 gallons 
by leakage and evaporation. Estimating the value of money at 6%, 
how much per gallon must he charge in order that he may realize 
the full amount of the cost, including the estimated interest ? 

8. If the consular fees collected by the United States consul at 
Liverpool in a year amount to £ 4,000, and his salary and expenses 
are $7,591.74, what percentage of the fees can be paid to the 
United States after deducting the salary and expenses, the esti- 
mated value of the pound sterling being $4.8665 ? 


APPENDIX IL. 


1.— A rule in arithmetic gives directions for performing the operations 
necessary to obtain a desired result. 

A clear understanding of subjects and principles will make rules unnecessary. 
The following, which refer to the more difficult processes, are given for reference. 
Quantities described concretely must often be considered as abstract. 


RULES FOR REFERENCE. 
2. — FACTORING. 


Greatest Common Divisor. I. (1) From the given numbers reject ail 
multiples of any of the given numbers. (2) Separate each of the remaining 
numbers into its prime factors. (3) Find the product of all the common prime 
factors. This will be the g. c. d. required. Or, — 

Il. Divide the greater number by the less and the preceding divisor by the 
last remainder till nothing remains. The last divisor is the g.c. d. If there 
are more than two numbers, find the g. c. d. of any two of them, and then of this 
g.c. d. and a third number, and so on. 


Least Common Multiple. I. (1) Reject fromthe given numbers all that 
are divisors of any of the rest. (2) Separate each of the remaining num- 
bers into its prime factors. (3) Multiply the largest of the given numbers by all 
the prime factors of the other numbers that are not found among its own. This 
product will be the l.c. m. required. Or, — 

Il. (1) Strike out any of the given numbers that are factors of any of the others, 
and divide the remaining numbers by any prime factor common to two or more 
of them. (2) Strike out from the resulting quotients and undivided numbers all 
that are factors of any of the rest, and divide as before. (3) Thus proceed until 
no two of the remaining numbers have a common factor. The product of the 
divisors and remaining numbers will be the 1. ¢. m. required. 

l 


2 APPENDIX I. 


3. — FRACTIONS. 


To Lowest Terms. Divide both terms of the fraction by their g. c. d. 


To a Required Denominator. (1) Divide the required denominator by the 
given denominator. (2) Multiply each term of the given fraction by the quotient. 


To Mixed Numbers. Divide the numerator by the denoninator. 


Mixed Numbers to Improper Fractions. Multiply the integer by the 
denominator, and to the product add the numerator. Write the sum over the 
“denominator. 


To a Common Denominator. (1) Find the l. c.m. of the denominators of 
the given fractions. (2) Divide this by the denominator of each fraction and 
multiply both terms by the quotient. 


Addition. (1) Change the fractions to like fractions with a common denom- 
inator. (2) Find the sum of the numerators. (8) Simplify the result. 

Add integers and fractions separately in adding mixed numbers. 

Subtraction. Proceed as in addition of fractions, excepting that the dif- 
ference of the numerators ts to be taken instead of their sum. 


Multiplication. I. When one factor is an integer. Hither (a) multiply the 
numerator by the integer, or (b) divide the denominator by it. 

II. When one factor is an integer and the other a mixed number. (1) Mul- 
tiply the integer jirst by the fractional part, then by the integral part of the multi- 
plier. (2) Add the partial products. 

III. When the factors are mixed numbers or fractions. (1) Change integers 
and mixed numbers to improper fractions. (2) Write the product of the numer- 
ators over that of the denominators. (3) Simplify the result. (The process 
may often be shortened by cancellation.) 


Division. I. A fraction by an integer. Hither (a) divide the numerator or 
(b) multiply the denominator by the integer. 

In general. II. (1) Change dividend and divisor to fractional form, and 
then (2) to a common denominator. (3) Write the quotient of the numerators 
over the common denominator ; or, III. Multiply the dividend by the divisor 
inverted. 


4.— DECIMALS. 
To Common Fractions. xpress the denominator and change the fraction 
to smallest terms. 


Common Fractions to Decimals. Annex decimal ciphers to the numerator 
and divide by the denominator. 


APPENDIX TI. 


co 


Addition and Subtraction. As with integers. 


Multiplication. (1) As with integers. (2) Give the product as many deci- 
mal places as there are in all its factors. 


Division. (1) Make the divisor an integer by removing the decimal point. 
(2) Move the decimal point in the dividend an equal number of places in the 
same direction. (5) Divide as with integers. (4) Give the quotient as many 
decimal places as have been used in the dividend. 


MEASUREMENTS. 


Nore. — In the following rules dimensions are spoken of as abstract numbers. 


5.— Or Lives. 
To find — 


The Perimeter of a Polygon. Take the sum of its bounding lines. 


One Dimension of a Parallelogram. JDivide its area by the given 
dimension. 


One Dimension of a Rectangular Prism. Divide its volume by the prod- 
uct of the two given dimensions. 


The Side of a Square. Jake the square root of its area. 

The Diagonal of a Square. Take the square root of twice its area. 

The Diagonal of a Rectangular Prism. Take the syuare root of the sum 
of the squares of its three dimensions. 


The Circumference of a Circle or Sphere. Multiply the diameter by 
3.1416. 


The Diameter of a Circle or Sphere. Multiply the circumference by 
0.31831, or divide by 3.1416. 


An Hypotenuse of a Right Triangle. Take the square root of the sum of 
the squares of the short sides of the triangle. 

One of the Short Sides of a Right Triangle. Take the square root of the 
difference of the squares of the given sides. 


6.— Or Surraces. 
Nore. — Before multiplying, dimensions must be changed to like numbers. 
To find the Area of — 


A Parallelogram. Find the product of base and altitude. 
A Triangle. Find half the product of base and altitude. 


4 APPENDIX I. 


A Trapezoid. Multiply the altitude by half the sum of tts parallel sides. 
Other Polygons. Divide into triangles and find the sum of their areas. 
A Sector. Find half the product of are and radius. 

A Circle. Multiply the square of the diameter by 0.7854. 


The Laterai Surface of a Prism or Cylinder. Multiply the perimeter of 
the base by the altitude. 


The Lateral Surface of a Pyramid or Cone. Multiply the perimeter of 
the base by half the slant height. 


A Sphere. Multiply circumference by diameter; or, multiply the square of 
the diameter by 3.1416. 


7.— Or Sots. 
To find the Volume of a — 
Rectangular Solid. Find the product of its three dimensions. 
Prism or Cylinder. Multiply the area of the base by the altitude. 
Pyramid or Cone. Multiply the area of the base by one-third the altitude. 
Sphere. Multiply the cube of the diameter by 0.5236. 


A Frustum of a Pyramid or Cone is what remains after cutting off the 
top in a plane parallel to the base. To find its volume: (1) Find the area of 
each of the two bases. (2) Add to their sum the square root of their product. 
(8) Multiply this result by one-third the altitude. 

To find — 


The Number of Board Feet in a Piece of Lumber. Multiply the prod- 
uct of its length and width in feet by its thickness in inches. (Disregard thick- 
ness when it is one inch or less.) 


8.— PERCENTAGE. 
To find the— 


Percentage. Multiply the base by the rate per cent. 
Base. Divide the percentage by the rate per cent. 
Rate per cent. Divide the percentage by the base. 


What constitutes base, percentage, rate%, etc., in business operations, is 
shown in the following table : — 


APPENDIX I. 5 
Rate ; 
BASE. PERCENTAGE. é AMOUNT. DIFFERENCE. 
PER CENT. 
£ Jos s Gain: | Cos SS ss: 
Profi ; Rate of Gain Cost plus Gain ;| Cost less Loss; 
and Cost. Gain or loss. Soraen or or 
Loss f: Selling Price. | Selling Price. 
: : A Rate of 
Insurance. Sum insured. Premium. —- -——- 
Insurance. 
Amount of 
Purchase eo Proceeds of 
a The Agent’s Rate of plus Commis- 
Commission. or 3 iF y aes : Sale less 
Commission. | Commission. sion; or ie 
Proceeds of ; Commission. 
Remittance. 
Sale. 
ace of Bill 
Trade hee ay en . Rate of Base less the 
. or The Discount. ; —— : 
Discount. ; F Discount. Discount. 
List Price. 
Valuation of | Total taxless | Shown by 
Taxes —- — 
all Property. poll-tax. tax on $1. 
Duties Cost as per Ad Valorem Rate of Cost of 
: Invoice. Duty. Tariff Importation. 


9.— INTEREST. 


General Method. Find the product of the principal, rate per cent, and 
time in years. (The time may be taken in months or in days, provided the 
product is divided by 12 or 360, as the case may be.) Use cancellation. 

Principal and interest added will give the amount. 


One Dollar Method. To find the interest of one dollar at 6%: Reckon 
6 cents for each year, 5 mills for each month, and + of a mill for each day. 

To find the interest of any sum: Multiply one dollar’s interest by the principal. 

For any rate except 6: Add to or subtract from the interest at 6% a propor- 
tional part of itself. 

Bankers’ Method. To find interest at 6%: Find 20 months’ interest by 
taking +5 of the principal, or 60 days’ interest by taking z)5 of tt. Then take 
such parts or multiples of these as the given time may require. 


6 APPENDIX I. 


Exact Interest. For any part of a year: Find the common interest for the 
exact number of days in the given part of a year; then lessen it by =|, of itself. 


> 


To find the Principal. Divide the interest by the product of the rate and 
the time in years. 


To find the Rate. Divide the interest by the product of the principal 
and the time in years. 

To find the Time. Divide the interest by the product of the principal 
and the rate per cent. The quotient is the time in 
years. 


To find the Amount due on a Note on which Partial Payments 
have been made. United States rule: Find the amount of the principal to 
the time when a payment or the sum of several payments shall equal or exceed the 
interest due at the time. Subtract such payment or sum of payments from the 
amount, and with the remainder as a new principal, proceed as before to the time 
of settlement. 


Compound Interest. Find the amount of the principal for the first period 
of time. Treat this amount as a new principal, and find its amount for the 
second period, and so on for the entire time. The last amount less the given 
principal will be the compound interest. 


Present Worth. Divide the given debt by the amount of $1 for the time to 
elapse before the debt is due. The debt less the present worth ts the true discount. 


10. — DISCOUNT. 


Bank Discount. Compute bank discount as tf it were simple interest on the 
Jace of the note for the term of discount. The face of the note less the bank 
discount will be the proceeds. 


True Discount. See Present Worth, § 9. 


11.— PROPORTION. 


Rule of Three. Make that number in the problem which is of the same kind 
as the desired result the third term of a proportion. 

If, from the conditions of the question, the result is to be larger than the third 
term, use the two like numbers in making the jirst ratio of the proportion less 
than 1; but if the result is to be smaller, make the first ratio greater than 1. 

Divide the product of the means by the given extreme, and the quotient will be 
the fourth term of the proportion, or the result desired. 


APPENDIX I. 7 
Partnership. (Give each partner such part of the whole gain or loss as his 
capital for any time is part of the whole capital for the same time. 


12. = ROOTS: 


Extracting the Square Root. 


I. Beginning at the decimal point, separate the given number into groups of 
two figures each. 

Il. Find the greatest square in the left group and place its root at the right ; 
subtract the square of this root from the left group, and to the remainder annex 
the next group for a dividend. 

Ill. Divide this dividend, omitting the last figure, by double the root already 
Sound, and annex the quotient to the root, and also to the divisor. 

IV. Multiply the divisor as it now stands by the last root figure and subtract 
the product from the dividend. 2 

V. If there are more groups to be brought down, proceed in the same manner 
as before. 


Extracting the Cube Root. 


I. Beginning at the decimal point, separate the given power into groups of 
three figures each. 

Il. Find the greatest cube in the left group and place tts root at the right. 
Subtract the cube of this root from the left group, and to the remainder annex the 
next group for a dividend. 

Ill. Annex a cipher to the root already found and take three times its square 
for a trial divisor. Divide the dividend by this trial divisor and place the quo- 
tient as the next root figure. 

IV. Multiply the number last squared by the last root figure and add three 
times this product and the square of the last root figure to the trial divisor for a 
complete divisor. 

V. Multiply the complete divisor by the last root figure, subtract the product 
From the dividend, and to the remainder annex a new group. 

VI. Form a second trial divisor, using two figures of the root with a cipher 
annexed, and proceed as before until all the groups have been used. 


13. — Roman Seven capital letters were used by the Romans to 
Notation. represent numbers. They are of almost no use in 
computations. 
Vv xX L C D M 


5 10 50 100 500 1000 


8 APPENDIX I. 


To represent other numbers, these letters are combined according to the 

following principles : 
I. Repeating I, X, C, or M, repeats its value. 
II. When I is used before V or X, X before C or L, and C before M, the 

difference of the values is to be taken. 

III. When any numeral follows one of greater value, a sum of values is to be 
taken. 

IV. A dash (—) over any numeral but I increases its value 1000 times. 


Show how these principles are illustrated in CC; IX; LX; CM; MM; C; 
MDCCCXCV. Mention four uses of Roman numerals. 


14.— Least Com- The least common multiple of two or more numbers 
mon Multiple. contains only such prime factors as are needed to pro- 
duce each number. 
The following method of finding the l.c. m. is a useful one, though not differ- 
ent in principle from that given on page 61. 


To find the l.c.m. of 24, 40, 72, 108. 


2.124  40'..72> 108 EXxpLaNaTIon. — We discard 24, for 

20 ~=.36 54 any multiple of 72 is a multiple of 24. 

10 18 27 We then divide the remaining numbers 

5s) Cae by any prime factor common to any two 

7 ie ar ee of them, until quotients are obtained that 

5 1 eS are prime to each other. The product 
2x2x2x3x3x5x3=1080, l.c.m. of the divisors and the remaining quo- 


tients is the desired least common multi- 
ple. Select from the process shown above the prime factors of each number. 


15.— Leap Years. A True or Solar Year is the exact time in 
which the earth revolves once around the sun. Its 
length is 865 d. 5 h. 48 min. 49.7 sec., or about 11} min. less than 365} days. 
To avoid the confusion and inaccuracy of the methods of reckoning time 
then in use, the Roman dictator, Julius Cesar, 46 B.c., reformed the calendar 
by establishing what is now known as the Julian year, of 3651 days. To avoid 
the inconvenience of counting the fractional part of a day every year, he 
decreed that three successive years should consist of 365 days and the fourth 
year of 366 days, the extra day being added to the month of February. The 
year containing the extra day is called bissextile or leap year. 
But this arrangement of the calendar made the civil year too long by about 
114 minutes, an error that amounted to 1 day in about 130 years. To correct 
this and other errors, Pope Gregory XIII. struck out ten days from the calen- 


APPENDIX I. > 9 


dar, calling Oct. 5, 1582, Oct. 15; and ordaining that thereafter only those 
centennial years should be leap years whose numbers are divisible by 400. 

The Gregorian year is now the civil or legal year in nearly all civilized coun- 
tries but Russia and Greece, where the Julian calendar is still in use, and the 
dates 12 days behind ours. 

The Gregorian calendar was ‘not adopted in Great Britain till 1752. The 
error had then amounted to 11 days, and hence the third of September was 
called the fourteenth. Old style dates are according to the Julian calendar, new 
style dates conform to the Gregorian calendar. 

When the number of a year ts divisible by 4, it is a leap year; but centennial 
years whose number is not divisible by 400 are exceptions. 


16.— Land Measure- Government Lands are divided by parallels 
and meridians into townships six miles square, 
containing 36 sections or square miles. Each 
section is divided into half sections and quarter sections. 

A township is designated by its number north or south of a base line running 
east and west, and east or west of a principal meridian running north and south. 

Thus, C is Township 4 N., Range 3 FE. Whatis A? B? 

The 36 sections into which a township is divided are numbered as in the 
accompanying figure. Point out section 15. 

Half and quarter sections are designated as W. or N. half sections, etc. ; and 
S. W. or N. E. quarter sections, etc. 


ments. 


Township. Section 15. 


Surveyors generally use, in measuring land, a steel chain 100 ft. long, 
divided into foot links, or a steel tape line of the same length graduated to feet 
and tenths. Sometimes a Gunter’s Chain is used. It contains 100 links, 
each 7.92 in. long. The chain is 4 rods, or 66 ft., or 792 in., in length. 80 
chains, or 320 rods, measure a mile. 


10 APPENDIX I. 


17.—The Metric The Metric System of weights and measures is 
System of Weights named from the Meter, from which all the units of 3 
length, surface, volume, capacity, and weight are 
derived. 

The Meter is approximately one ten-millionth of the distance from equator 
to pole on the earth’s surface. 


and Measures. 


Notr. — The Metric System is in general use by nearly all civilized nations except Great Britain 
and the United States. It is used by some departments of the United States government, and in 
the sciences. 

The Metric System is a decimal system, ten units of one denomination making 
one of the next higher. 

Decimal parts of the standard units are denoted by Latin prefixes ; multiples 
of the standard, by Greek prefixes. 


Milli means 1000th Myria means 10000 
Centi means 100th Kilo means 1000 
Dect means 10th Hekto means 100 


Deka means 10 
In the tables units in common use are in étalics. 


Length Measures. Standard unit, the Meter. 


Table. Equivalents. 
10 millimeters (™™) =1 centimeter (™)  =0.8937079 inch 
10 centimeters = 1 decimeter (#™) 
10 decimeters = aecer (=) = 9.57079 inches 
10 meters = 1 dekameter (P™) 
10 dekameters = 1 hektometer (#™) 
10 hektometers = 1 kilometer (=) tees 


~ (0.621382 miles 
10 kilometers = 1 myriameter (™™) 


Surface Measures. Principal unit, the Square Meter. 


Norr. — As the units of surfaces are squares whose dimensions are the corresponding linear units, 
it takes 10? or 100 units of one denomination to make one of the next higher. 


Table. Equivalents. 
100 sq. millimeters (84™™) = 1 sq. centimeter (9ae™) = 0.155 sq. inch 
100 sq. centimeters = 1 sq. decimeter (#1 ¢™) 
~ 100 sq. decimeters = 1 sq. meter (°4™) = 10.764 sq. feet 
100 sq. meters = 1 sq. dekameter (#1 Dm) 
100 sq. dekameters = 1 sq. hektometer (#1 Hm) 
100 sq. hektometers = 1 sq. kilometer (25™) = 247.114 acres 


Nore. — When used in measuring land the square meter is called a centare (ca), the square deka- 
meter an are (a), and the square hektometer a hektare (Ha), 


APPENDIX I. heh 


Volume Measures. Principal unit, the Cubic Meter. 


Norr. — As the units of volume are cubes who edges are the corresponding linear units, it takes 
108 or 1000 units of one denomination to make one of the next higher. 


Table. Equivalents. 
1000 cu. millimeters (2 ™™) = 1 cu. centimeter (C%e™) =,0.06103 cu. inch 
1000 cu. centimeters = 1 cu. decimeter (cu am) 
1000 cu. decimeters = 1 cu. meter (c%™) = 35.314 cu. feet 


Nore. — In measuring wood the cubic meter is called a Stee (18t = 0.2759 ed.) ; a decistere (1%*) 
is one tenth of a stere. 


Measures of Capacity. Principal unit, the Liter = a cu. decimeter. 


Table. Equivalents. 
10 milliliters (™) = 1 centiliter (*!) = 0.6102 cu. inch 
10 centiliters == 1 deciliter (4!) 

oe : 1.0567 liquid quarts 

— ] — 
10 deciliters = 1 liter {*) = { 0,908 diy quart 
10 ljters = 1 dekaliter (DP!) 

26.417 gallons 
= Hl) — 5 

10 dekaliters = 1 hektoliter (™) = 2.8375 bushels 
10 hektoliters = 1 kiloliter *) 


Notre. —The dizer is used in measuring liquids and small fruits, the hektoliter in measuring 
grain, vegetables, and liquids in larger quantities. 


Measures of Weight. Principal unit, the Gram. 


Table. Equivalents. 
10 milligrams (™8) = 1 centigram (°) = 0.15482 grain 
10 centigrams = 1 decigram (28) 
10 decigrams = 1 gram (8) = 165.432 grains 
10 grams =1dekagram (2) 
10 dekagrams = 1 hektogram (8) 
10 hektograms =1 kilogram (®8) = 2.20462 pounds 
10 Kilograms = 1 myriagram (M8) 
10 Myriagrams =1 quintal (&) 
10 Quintals =1 metric ton(T) = 2204.621 pounds 


Nore. — The gram is the weight of a cubie centimeter, the kilogram of a cubic decimeter, and 
the metric ton of a cubic meter of distilled water at its greatest density. 

The gram is used in mixing medicines, and in weighing jewels, precious metals, letters, ete. 
Ordinary articles are weighed by the kilogram (commonly called ilo), and heavy articles by the 
metric ton, 


12 f APPENDIX I. 


18.— TABLE OF EQUIVALENTS. 


Common. Metric. Common. Metric. 
1 inch = 2.54em 1 cu. foot = 28.31 7cu dm 
1 foot = 30.48em leu. yard =0.7645cum 
1 yard = 0.9144™ 1 cord = 3.6248 
1 rod = 5.029m 1 liquid quart = 0.94631 
1 mile = 1.6093Km 1 gallon = 3.785! 
1lsq. inch = 6.452sacm Ldry quart: #11013 
1 sq. foot = 9.290384 dm 1 bushel = 0.3524H1 
1sq. yard =0.8361sam 1 grain = 0.06488 
1 sq. rod = 0.25298 1 ton = 0.9072met ton 
1sq. mile =2.59saKm 1 troy ounce = 31.10358 
1 Acre = 0.4047Ha lav. ounce = 28.358 
leu. inch =16,387cucm lav. pound = 0.45386Ks 


Approximate Equivalents. 


1 decimeter = 4 inches 1 liter = 1.06 liq. qt. or 5%, dry qt. 
1 meter = 3 ft. 33 in. 1 dekaliter = 1 pecks 

1 dekameter = 2 rods 1 hektoliter = 22 bushels 

1 kilometer = $ mile 1 gram = 15} grains 

1 acre = 4 sq. rds. 1 kilogram = 21 av. pounds 
lhektare = 23 acres 1 metric ton = 2200 pounds 

1 stere = } cord 


19.— Metric System. ‘The units of the Metric System form a decimal 
system. Hence the following principles apply : — 

I. Excepting in square and cubic measures, any 
metric number may be changed from one denomination to the next smaller. or the 
next larger by moving the decimal point one place to the right or left, as the 
case may be. 


Written Exercises. 


II. Jn square or surface measures this reduction is effected by moving the 
point two places, and in cubic or volume measures three places, instead of one. 
e 


Ill. Any denomination may be taken as the unit, the number at the right of 
the point being read as its decimal. 
Explain the following changes or reductions : — 


1, 3247.28™ = 324728 = 32,4728Hm — 3,24728Km — 3247280mm, 
9, 67317.9694m = 673.1796s4 dm — 6,73179624 m = 0.0673179684 Dm, 


APPENDIX I. 13 


8, 8.3724H8 — 837.249 — 83724¢4 = 8372454 ™, 

4, 47.2384cum — 47,234st — 47234cu dm — 47234000cu em, 

5, 247.831! = 2.47831H! — 24783. 141. 

6, 1846.982% = 1.346982K = 134698.2¢s. 

7, In 847.2K, how many grams? How many pounds? 

8, Change 75 bushels to hektoliters. 

9, How many square meters in a rectangle 18 ft. by 10 ft. ? 

10, An importer pays duty on 1,200 meters of cloth. How many yards ? 
11, How many square rods in a square hektometer ? | 

12, How many liters in a cubic meter ? 


18, An importer buys 250! of liquor at $0.75 a liter. He sells it for $3a 
gallon. What does he gain or lose ? 


14, A rectangular stone is 1™ long, 54™ wide, and 24 thick. How many 
kilograms does it weigh, being eight times heavier than water ? 


15, How many kilograms of flour in a barrel ? 
16, Add 18.32K™, 648m, 94.8Hm, 38 ,4dm, 
17. What will a stere of wood cost at $12 a cord ? 


18, How many hektares in a field 144™ long and 40D™ wide? How many 
acres ? 


19, How many gallons in a cubic meter of water ? 
90, How many times is 164™ contained in 1,.28K™ ? 


91, If goods are bought at $2.55 per yard, at what price per meter must they 
be sold to gain 25%? (1 meter = 39.37 inches.) 


99.. A hektoliter of fruit weighs 63 kilograms, and 32 liters of syrup can be 
obtained from it. How many kilograms of fruit will it take to make a hekto- 
liter of syrup ? 


93, The distance between two places on a map is 12.5 centimeters. What is 
the actual distance between the places if the scale of the map is 1 to 60,000 ? 


94, If a certain stone is 2.83 times as heavy as water, what is the weight 
of a piece of this stone which is 5.59™ long, 17.364" wide, and 52.6e™ thick ? 


14 APPENDIX I. 


20. — Comrounp InTEREST TABLE. 


vay 2 per cent. | 24 per cent. | 3 per cent. |34 per cent. | 4 per cent. | 5 per cent. | 6 per cent. 


1 1.020000 1.025000 1.030000 1.035000 1.040000 1.050000 1.060000 
2 1.040400 1.050625 1.060900 1.071225 1.081600 1.102500 1.123600 
3 1.061208 1.076891 1.092727 1.108718 1.124864 1.157625 1.191016 
4 1.082432 1.103813 1.125509 1.147523 1.169859 1.215506 1.262477 
5 1.104081 1.181408 1.159274 1.187686 1.216653 1.276282 1.338226 


6 1.126162 1.159693 1.194052 1.229255 1.265319 1.340096 1.418519 
7 1.148686 1.188686 1.229874 1.272279 1.815932 1.407100 1.503630 
8 1.171660 1.218403 1.266770 1.316809 1.868569 1.477455 1.593848 
9 1.195093 1.248863 1.304773 1.362897 1.423312 1.551828 1.689479 
0 1.218994 1.280085 1.848916 1.410599 1.480244 1.628895 1.790848 


11 1.248374 1.312087 1.384234 1.459970 1.539454 1.710339 1.898299 
12 1.268242 1.344889 1.425761 1.511069 1.601032 1.795856 2.012197 
~ a: 1.293607 1.878511 1.468534 1.563956 1.665074 1.885649 2.182928 
14 1.319479 1.412974 1.512590 1.618695 1.731676 1.979932 2.260904 
15 1.345868 1.448298 1.557967 1.675349 1.800944 2.078928 2.896558 


16 1.372786 1.484506 1.604706 1.733986 1.872981 2.182875 2.540352 
aT 1.400241 1.521618 1.652848 1.794676 1.947901 2.292018 2.692773 
18 1.428246 1.559659 1.702433 1.857489 2.025817 2.406619 2.854339 
19 1.456811 1.598650 1.753506 1.922501 2.106849 2.526950 3.025600 
20 1.485947 1.638616 1.806111 1.989789 2.191123 2.653298 3.207136 


21.— Annual Interest. 1, I borrow $800, agreeing to pay 6% interest 
at the end of each year. I do not, however, pay 
any interest until the end of 4 years 8 months, when I pay principal and interest. 
What should I then pay ? 
In cases where there is an agreement to pay interest annually the custom is 
to charge simple interest on the principal and on the overdue interest for the 
time each interest payment is overdue. ‘This is called annual interest. 


Process. 


Each payment of annual interest should have been $48. 

The 1st annual interest payment of $48 has been due 3 yr. 4 mo. 

The 2d annual interest payment of $48 has been due 2 yr. 4 mo. 

The 3d annual interest payment of $48 has been due 1 yr. 4 mo. 
In all, interest on $48 has been due for . . . Tyr. 


APPENDIX I. 


Interest of $800 for 4 yrs. 3 mo. . 


7 yr. interest of $48 
Total interest due 
Principal due 


Amount due at arrieaane 


Find the annual interest of — 


9, 31,200 for 4 yr. 6 mo. 


3, 1,800 for 5 yr. 8 mo. 


4, 4,200 for 3 yr. 2 mo. 


5, 640 for 5 yr. 7 mo. 


at 6% 
at 5% 
nite 


at 5% 


co co “I OD 


15 


ae Mere ae ye $ 204 
20.16 
$ 224.16 

800 
$ 1024.16 


$900 for 9 yr. 6 mo. at 7 ¥, 

720 for 3 yr. 8 mo. 12 d. at 44% 
. 618 for 4 yr.5 mo. 17d. at 3% 
, 427 for 6yr.8 mo. 2d. at 12% 


22.— Taste sHowinc RATE oF INTEREST ALLOWED IN THE STATES AND 


TERRITORIES. 


The legal rate is given in the first column. 


in the second column. * Any rate. 


upwards on collateral security. 


States. ape 
per cent. 
BAIR DATING ceietais acre Selec 
A YIZOUS ed cas ote ota 7 * 
TATEKSNIBAR S Aotaticteye noe 6 | 10 
Californidentesncec oes vi * 
COlOTAAO: © a4 oes carne ae S| * 
Connecticut.......... 6 
Dela wareiis secrete 6 6 
Dist. of Columbia....| 6] 10 
OEIC A ccc itatt cee taal ig 
EOTTAseldtis «os owen es q 8 
TdANO scene < one cites eb ats: 
linotsisaece ean os ele a 
Tridiana.n.2 ss eee e 6 8 
LOWS eine here 6) 8 
TCANBBSE. wees at 6 | 10 
IONTUCE Yat ass 6 6 
} 


States eee 
per cent. 
OUisiaN Sass eo secs 5 8 
MERIT OP a ocr tore eyes ere arr 6 * 
Maryland. swe ofecacs's 6 6 
Massachusetts....... 6 ma 
MICHIE ST precepts ee sais 6 8 
IVETTAT ESO US teietstete stele, ss ed ady) 
Mississippi pew fees 6 | 10 
Missoutine- aeons. 6 8 
IM OTICATIONS amtsye- cosy: 10 * 
INGDTASKie conics «assis 7 | 10 
IN GY Ad Saee was xe ies q * 
New Hampshire..... 6 6 
New Jersey ......... 6 6 
New Mexico:........ 6 | 12 
IN GW X Of katstetean «tes 6 6 
North Carolina ...... 6 8 


North Dakota >..<s.. q 


The rate allowed by contract is 


_ 
bo 


+ Any rate on call loans of $5000 and 


States. ae 

: per cent, 
CIO Rr ates ve thae ee oe 6| 8 
Okishorm a esckee cee oe 7 9 
OVEPOM A sen see 4.0 « 8 | 10 
Pennsylvania ........ a ake 
Rhode Istandiccns..... 6 * 
South Carolina...... (i 8 
South Dakota... 2. T | 12 
"ROTINEBSEC is 6.5 ope eae 6 6 
PEXASi. Say as ete eect. 6 | 10 
Ginter eree: 8 * 
VierMlOnitimad: ore «wep 6 6 
WV APU Reaver sets eee Oda Me 
Washington -2.-.=0- olan | Ba 
Wiest. Vinpinignss:..2-|) Ol) 
WiSCOnBII;eMesenaees |" Clo LO 
VVEY OMEN Ys rcnetaiey islet 1a 

i 


i6 APPENDIX I. 


23.— Partial Pay- When a note is settled within a year from its date, 
it is the custom of business men to use the following 
method : — 

Find the amount of the face of the note from its 
date to the time of settlement. 

Find the amount of each payment from its date to 
the time of settlement. Add their amounts and take the sum from the amount of 
the note. The remainder is what is due. 


1, A note for $1200 dated May 7, 1895, is settled April 25, 1896. What is due 
at settlement if it draws 6% interest and has the following indorsements ? 


Aug. 19, 1895, $400. Oct. 25, 1895, $300. Jan. 25, 1896, $450. 


ments of Promissory 
Notes. Merchants’ 
Method. 


Process. 
Amount of $1200 from 5/7, ’95 to 4/25, 96, 11 mo. 18s 4 Se ee 1 26000 
Amt. of $400 from 8/19, 795 to 4/25, ’96, 8 mo. 6 d. . . $416.40 
6 66) 3300 10/25, °95 to 4/25, °96,4mo. . .. . 306. 
8 eae A EA) 1/25, °96 to 4/25, °96,3mo. . . . . 456.75 
Total-amountof payments "= es ee 
Amount due at settlement Sammie, ee G $90.45 


9, An 8% note for $1500 dated Aug. 4, 1896, is settled June 28, 1897. $460 
was paid Oct. 4, 1896 ; $500, Jan, 16, 1897. What is due at settlement ? 

3, Face of note $6000 ; its date May 15, 1897; rate 4%. $2000 paid Aug. 20, 
1897, and $3000 May 1, 1897. What is due at settlement, May 9, 1897 ? 


24.— Concerning Promissory Notes, étc. 1, A Joint Note is one 
which two or more persons jointly promise to pay, each one of whom must put 
his signature to the note and is held liable for his share and no more. 

9, A Joint and Several Note is one which two or more persons jointly and 
severally promise to pay, each of whom must put his signature to the note and is 
held liable for the whole amount provided others fail to pay their share. 

3, Notes written in pencil are valid, but should be written in ink. 

4, When the words and figures expressing the face of a note or check, or 
draft, differ, the words govern the amount to be paid. 

5, Unless otherwise stated a note is payable at the maker’s place of business 
or residence. 

6, Notes cannot be collected before maturity. 

7, Demand notes are due and payable immediately on presentation. They 
bear legal interest in all cases after demand for payment is made. 


APPENDIX I. 17 


8, In most States a note becomes outlawed or worthless at the expiration of 
six years from its maturity, or from the date of the last payment of either prin- 
cipal or interest. 

9, Notes are void if given on Sunday, or by a minor, or without considera- 
tion, or if obtained by force or illegally. 

10, Grace is not generally allowed on demand notes nor on sight drafts. 

11, Bankers throughout the country are seeking to secure the abolishment of 
days of grace on all commercial paper by action of State legislatures. Hence 
changes are likely to occur in States now allowing grace. 

12, Business men generally date their notes so that they shall not fall due on 
Sunday or on a legal holiday. A note falling due on Sunday, or on a legal holi- 
day, is in most States payable on the business day next preceding. In the fol- 
lowing named States and Territories it is payable on the next business day 
succeeding : — 


Alabama Idaho Nebraska Pennsylvania 
Arizona Louisiana New Jersey Rhode Island 
California Massachusetts New Mexico South Dakota 
Connecticut Michigan New York Vermont 
Dist. of Columbia Mississippi North Dakota Wisconsin 
Florida Missouri * Oregon 


25. — Equation of 1, Show that the interest of $200 for 4 mo. is the 
same as the interest of $800 for 1 mo. or of $1 for 
800 months. 

9, May 10, a debtor owes his creditor $200, due without interest in 3 months, 
$600 in 4 months, and $500 in 6 months. When may the $1300 of indebted- 
ness be paid at one time without loss of interest to either debtor or creditor ? 


Payments. 


Process. 
( $200 for 8 mo., or of $1 for 600 months. 
The debtor is entitled to the use of ~ 600 for 4 mo., or of 1 for 2400 months. 
500 for 6 mo., or of 1 for 38000 months. 


The debtor is entitled to the use of $1 for a total of 6000 months. 


This is the same as the use of $1500 for 3155 of 6000 mo., or for 4,5, months, 
or 4 mo. 17 d. 

4 mo. 17 d. after May 10 is Sept. 27, the date on which a single payment of 
$1300 may be equitably made. This date is the average or equated time of 
payment, and the process of finding it is the Hquation of Payments. 


* Unless a Sunday or holiday follows. 


18 , APPENDIX I. 


8, Find the equated time of payment of $100 due in 2 mo., $200 due in 
38 mo., and $3800 in 4 mo. 

4, When may these debts be equitably paid: $550 tet in 60 days, $400 due 
in 90 days, $700 due in 30 days ? 


5, Aug. 10, $320 is due; Sept. 15, $424.75; Dec. 4, $612.40. What is the 
equated date of payment ? 


Process. 


Aug. 10, $320 is due. 

Sept. 15, or 86 days after Aug. 10, $425 * is due. 
Dec. 4, or 116 days after Aug. 10, $612* is due. 
320 x 0 days = 0 days 
425 x 36 days = 15,300 days 
G12 x 116 days = 70,992 992 days 

1357) 86,292 days (64 d. 


64 days after Aug. 10 is Oct. 13, the equated time. 


6. $500 is due March 31, $600 May 1, and $400 June 1. On what date 
may a single payment of $1,500 be made ? 


7, Brown & Fisher owe Chandler & Co., $350, which is to be paid July 10, 
$62 due Aug. 1, $108.75 due Sept. 10. What is the equated date of payment ? 
Find the equated time of payment for these obligations : — 
8, $800 due May 15; $650 due June 11; $390 due Aug. 4. 
9, $397.25 due Nov. 1, 1895 ; $64.90 due Dec. 10, 1895 ; $480 due Feb. 1, 1896. 


10. I hold three non-interest-bearing notes: one for $86.40 maturing June 
7, one for $348.96 due Aug. 5, and a third payable Sept. 16 for $408. I 
exchange them for a single note maturing at the equated time of payment. 
Required its face, time to run, and maturity, without grace. 

Nor. — If there is a common term of credit, we may disregard it in finding the equated time, and 
add it to that time when found. 

11, There is a common term of credit of 2 months on the following: $600 
bought Oct. 5, and $500 bought Dec. 9. When may both be paid at once ? 

12. May 1, Samuel Rich bought a bill of $375 worth of merchandise, July 10 
he bought $480 worth, and on Aug. 22, $218.75. On each purchase he was to 
have 60 days’ time. What is the equated time of settlement ? 


* Reckoning to the nearest dollar. 


APPENDIX I. 19 


26. — Average of 1, The following is a statement from the books 
of Bainbridge & Co., of their account with William 
Henderson. We wish to find the average date or 
equated time for the settlement of the account without loss of credit or interest 


to either party. 


Accounts. 


WILLIAM HENDERSON. 


Dr. Cr. 
1896. 1896, 
Mar. 20 | Mdse. 30 d. $400 Apr. 7 Mdse. 2 mo. $300 
Apr. 5 a 2 mo. 500 May 5 Note 4 mo. 400 
June 12 SOU CEs 800 
Process. 
Debits. Credits. 
Term Term | - 
Date. of Due. | Debt.| Days.| Product. | Date. of Due. |Credit.| Days.| Product. 
Credit. Credit 
Mar. 20 | 30d. | Apr.19| $400] 19 7600 | Apr. 7] 2 mo. | June 7 | $300 68 20400 
Apr. 5| 2mo.| June 5} 500] 66 83000 | May {7 |4mo.!Sept.5} 400 | 158 63200 
June 12} 60d. | Aug. 11] 800] 133 106400 Te 
———_ $700 - 83600 
$ 1700 147000 
700 83600 
$ 1000) 63400 (63 days after Mar, 31= June 2. 


We find when each item of the debits and credits becomes due by adding its 
term of credit to its date. 

We assume March 31, the last day of the month preceding the earliest date 
at which any item falls due on either side of the account, as the focal date. 

We then multiply each item by the number of days between this focal date 
and the date of its maturity, and find the sum of the items and the sum of the 
products on each side of the account. 

The aggregate time credit of the debit items is $1 for 147,000 days. That of 
the credit items is $1 for 83,600. 

This is an excess of $1 for 63,400 days in time credits on the debit side of the 
account. But the balance of the account is $1000, and $1 for 63,400 days = $1000 
for 63 days, which is the average term of credit for the balance due, and is to be 
reckoned forward from March 31, the focal date. 

This gives June 2 as the date when the balance is equitably due. 


20 APPENDIX I. 


Nore. — When the balance of items and the balance of credits are on different 
sides of the account, the average term of credit comes before the focal date. 


Find the equated time for paying the balance of these accounts : — 


Dir: Cr. 
9, Jan. 4, Mdse 3 mo, $600 Mar. 1, Mdse 2 mo. $500 
Heb. 10, eke. 110, 700 ADT at eno: 200 
Dre Cr. 
3, Mar. 17, Mdse 60 d. $425 Jan. 24, Mdse 2 mo. $ 900 
Apr. i; Sis 600 
May, tives) o1n0; 783 
Dr Cr. 
4, Aug. 15, Mdse 4 mo. $725 May 11, Mdse 3 mo. $611.84 
LV nL Oy ares em OO le 500. 
DODi dene. an Oud: 317.16 
Or, Cy 
5, Sept. 15, Mdse 60 d. $215 Aug. 19, Mdse 90d. $131.75 
Oct. 19, 5 OU a1; 641 Noy, 10° seonG, 250 
Dr. Cr. 
6, July 17, Mdse38mo. $1500 July 24, Cash $ 500 
Apne 17, "=e; Samo. 800 Aug. 19, Mdse 90 d. 400 
Sept. 24, <“* 2mo. 1000 Sept. 28, Note 30 d. 1500 
Sept. 24, ‘* 3 mo. 1500 Oct. 29, Cash 200 


27. — Extraction of 1, Find the value of 38; 30%; 300%; 9%; 908; 
the Cube Root. V27; V 27/000; ~/27/000/000; 53; W125; 503; 
V 125/000; 6003; V 216/000/000. 


9, Compare the number of places in the 3d power or cube with the number 
in the 3d or cube root. 


3, If the cube of a number contains three times as many places as the num- 
ber, or one or two less, how many places in the cube of 253? 67? 8000? 
4, How many figures in the cube root of 729 ? 512,000 ? 8,000,000 ? 


5, Cube 0.08. How many decimal places in the cube? Why? 6, How 
many decimal places in the cube of 0.716? Of any number? 7, Why must 
the number of decimal places in the cube of a decimal be a multiple of 3 ? 

The square of a number of two digits consists of three parts [§ 333]. So its 
cube is made up of four parts. Let us analyze the process of cubing a number 
to see what these four parts are. 


APPENDIX I. 21 


To find the cube of 64. 
643 = 64 x 64 x 64; or, as 64 = 604+ 4, 
648 = (60 + 4) x (60+ 4) x (604+ 4) = (¢4+ 0) (t+0)(t+o0). 
60+4=t +0 
60+4 =t +0 


60x 4 +4%=to +0? 
607+ 60x4 =? + to 


602 + 2(60 x 4) + 427 =@ +2to+ 0? 
60+4=¢t +0 


602 x 4+ 2(60 x 4?)+ 48 = #0 + 2to? + 08 


608 + 2(602 x 4) + 60x 4? =f +270 -+ to? 
603 + 3(602 x 4) + 8(60 x 44)4+ 48 = #8 +30 + 3t02 + 08 
216,000 + 43,200 + 2880 + 64= 262,144 


The four parts of which the cube of 64 is composed, are 


I. The cube of the tens. III. 3 times the tens times the ones?. 
II. 3 times the tens? times the ones. IV. The cube of the ones. 


Let us try to find the factors of these four parts while finding the cube root 
of 262,144. 


Process. 
8430+43t0e2+03 = 262/144 (60 +4. 
te — 216 000 
$70 = 10,800 | 46 144 = 30 + 3to? + 0’ = (84 3to + o*)o. 
3 i = pa EXPLANATION. — 262,144 comes between 
Ge 


ee the cubes 216,000 and 348,000 ; its cube 
387+3to+07= 11,536 | 46 144 root then comes between 60 and 70, and 
Part I. the ¢* must be 216,000. Taking this out of the cube, what remains ? 

Part IL, or 50 is much the largest of the three remaining parts of the cube. 
It is made up of two factors, 3¢2 and 0. The remainder, 46,144, is approxi- 
mately the product of these two factors. 38¢2=3 x 60?=10,800. If we divide 
the approximate product, 46,144, by one of its factors, 10,800, we get the other 
factor, 4 or o. 

Part Ill., or 8to*=3toxo0; S$to=3x60x4=720. Part IV., or of=07x0; 
o?=16; uniting 32+ 3to+ 0%, 10,800 + 720 + 16, we have 11,536; multi- 
plying this number by o or 4, we have 3 to? + 3to? + 0° = 46,144. Hence we 
conclude that 64 is the cube root required. 


29 APPENDIX I. 


28. — Exercises in As shown above find one of the three equal 
Cube Root. AM eats Si Brag 
1, 12,167. 3, 46,656. 5, 103,823. 
Written. 9. 32,768. 4, 74,088. . 6, 157,464. 
Lt ,01LOs 9, 373,248. 11, 614,125. 
8. 238,328. 10. 474,552. 12, 884,736. 


18, What is the edge of a cubic block containing 421,875 cubic inches ? 

14, How large a cube will weigh as much as a rectangular prism 24 in. long, 
16 in. wide, and 9 in. thick ? 

15, Extract the eube root of 320.013504. 


Process. 
320.'015/504 (6.84 
216 1. We begin at the decimal point to 
3 x 60? = 10,800] 104. 018 separate the power into groups of three 


3x60x8 = 1,440 
g2 — 64 

12,304 

3 x 6802 = 1,387,200 
3x 680x4= 8,160 
43 16 
1,395,376 


figures. How many figures will then be 
in the root? How many integral ? 
98. 482 2. Having found two figures of the 
5. 581,504 root, we consider the tens to be 68, and 
proceed as before. 


| 


3. If the number is an imperfect 
power, we may annex decimal ciphers 
and approximate the root. 


16. V3796416 18, V37.259704 90. 0.087640 
17, V12977875 19, V0.1001728 91, V25 


In extracting the cube root of fractions or mixed numbers, follow the direc- 
tions given under square root (page 221). 


5. 581,504 


22, Vit 94, V38 96, 724 
23, VitGts 25. Vath 27. V405 2% 


98, Find the entire surface of a cube whose volume is 37 cu. ft. 64 cu. in. 
99, Ifa cubical pile of wood contains 108 cords, how long is it ? 


29.— Foreign Moneys. The director of the United States Mint publishes 
from time to time a list of nearly forty foreign 

countries with the value of their money standards in United States gold coin. 
Many of these values change from year to year, especially in the case of coun- 
tries which have a silver standard, and the rates of exchange vary accordingly. 


APPENDIX II. 


INTRODUCTION TO ALGEBRA. 


1.— Likened to In arithmetic we use (1) numbers expressed 

Arithmetic. _in figures, (2) signs that indicate some process 

or shorten an expression, and we may use 

(3) letters that represent a number when we cannot or do not wish 
to give the size of it in figures. 

In algebra we use the same numbers, signs, and processes, and 
we constantly use letters to represent quantities both known and 
unknown. Ror 

Adding: 5+5= how many 5’s? a+a? 

Subtracting: 10—5? Twod’s—5? 2a—a? 

Multiplying: 4 x 5=how many 5’s? 4xa? 

Factoring: 20=? 4x5=? 2-2a? 4a? 

Dividing: 20+4? 20+5? 4%) 44 

a a 

Squaring: 5?= Lloro. xw.; @=? V25=? 


2.—Contrasted with The most important differences are 
Arithmetic. these : — 

I. In algebra, before solving a problem, 
it must usually be stated in one or more equations to show the rela- 
tion between the known quantities and the unknown. 

The chief thing to learn is how to find from equations the value 
of the unknown quantities which they contain. 
II. All algebraic problems involve the use of literal quantities. 
23 


24 APPENDIX II. 


[$$ 76-79 of the arithmetic may be reviewed here. ] 

Though arithmetic serves for ordinary computations, there is 
many a problem that cannot be easily solved without the equations 
in which it may be stated. 


3. — Letters and The first letters of the alphabet, a, b, c, d, 
Signs. etc., are used as abbreviations or convenient 
substitutes for quantities whose value is given 
or supposed to be known. 
The last letters, w, y, z, etc., are used for quantities of unknown 
value. 
Signs have the same force as in arithmetic. 


4.— Reading and Read the following expressions and say what 
Interpreting. each means. 
Thus, in S 


a the literal quantity @ is to be subtracted from ec, 
Cc _— 


and the numerical quantity 3 is to be divided by the remainder. 


1. 7+5b 4. g+T7 7, 2 9 3 
Orie 0 5. f+8—ad - oat 
3. 3—C¢ 6. dxa+2 ea ewe 


11. Which of the preceding expressions have the fractional form? 
Read them as if the sign + had been used. 


Notice that 2 abc is read ‘‘Twoa, 0, c’’; that it means the same as2-a+-b-cor2xax0xe, 
and is the common way of indicating the product of such factors, the numerical factor coming first. 


Read and interpret the following. Notice which are the unknown 
quantities. 


Thus: 5a@ indicates the product of the numerical factor 5 and of the literal factors @ and @. 


12. 6def 15. 25 mn 18. 4ab+ 3bc+ d5cd —~- 
1322 ae Ae 16. ay —32+10 19. atb 
14. 5abm + 2cd 17. Tadmay cd 

20. 34—2 91, CEe+3my 


z ben 


APPENDIX II. 25 


5.—FProblems Sug- 1. Make a problem of which the proper 
gested by Equations. statement is 3 x $12=x. What is meant by 
oe? kts. value isl: 


Try to make w problem for each of these statements : — 


2. «=13—4 4. $17 -—aw=$10 6. 4wk. =32wk.—2 
& #2—-12h=8h 56 5xa= ts (pee bu = oe DU 
“a ii ss 
8. $1—a2f = $34 $4, 9. $34 94-24) 
Ai 
6. — Writing Equa- The first member of an equation, the part 
tions. at the left of the sign =, must always equal 


the second member, the part at the right. 


Put into the form of equations, and test each one: — 

10 added to 35 is the same as 9 times 5. 

3 more than half of 7 is like a quarter of 26. 

15 = twice 10, less 2 more than 3. 

What quantity = 19 less 5 times the double of 14? 

3 =a half as much as 1 more than 10 from 15. 

a from a leaves 1 less than a fifth of five. 

5 = as much more than 4 as 3 exceeds 2. 

A quarter of a is as much as the whole of b. a=44; b=11. 


ee en ee Oe ie 


4 less than a and 6 = 1 more than } a hundred. 


a less 11 comes within 3 less than 70 of being 100. 


= 
Oo 


7.— Expression of 1. One chimney has « flues; another twice 
Unknown Quantities. as many, or 2a; a third has as many as the 
first two, or .. In all how many ? 
2. y is a certain number. One farm employs 3y workmen, 
another 4 times as many, or .. The two together 


296 APPENDIX IL. 


ae x iF 
3. Compare the values of # and ms of 5 and «; of 31a and 


8a v 
aaa 


4. One kind of pillow costs 3 times a certain sum, another 8 times 
as much as the dirst, or 22%. 
_ 6. One ladder contains 2 rounds; another 4 as many, or —; 
another 8 more than the first or ..; another contains 2 more than the 
second ladder. All four contain 

6. If 16a will pay for 4 tons, express the cost of 4 of a ton as 


more or less than a. 
22 


9 
o 


8. Explain the difference between y — 5 +7 and y—(5 +7). 
Each expression is how much more or less than ¥?  [§ 75.] 
9. How many @’s added will equal # x (5 — 2) or (5 — 2)@. 
10. One shelf in a case holds within 1 of a quarter of all the books, 


7. Compare 58 of # with 2a; with 4 of 2a; -with 


TED 


a 
or ——.; another holds a quarter as much as 4 more than the 


Sal 
‘ ‘ 


whole, or = aloes 11. There are - ot a OF _ on the third shelf, 


Ja |= 
5 ven 5 @ 
and on the fourth Peano ae of « = —- 
d — 
12. «=the value of an apple; 6 peaches cost 12a, 4 plums 62. 


How many plums have the value of 9 peaches ? 


—n 


8.— Statement of I. Tell what unknown quantity is to be 


Problems. found. 
II. Make an equation for each case. 


III. Then give # or y its numerical value. 
1. A bag contained w coins. Each of 8 persons took out 7, and 
4 remained. 


2. When six marbles had been lost from a bag containing y, each 
of 4 persons took out 8 and 3, leaving none. 


APPENDIX II. iy 


8. The present yield of apples (7) is 10 barrels more than last 


OR 


year’s, when the crop bought $160 at $2.50 a barrel. 


4. A certain bicycle cost $y. One half that amount is the same 
as 3 the cost of a $75 bicycle. 
2 4 . . : Ai 
5. If you take 1 of w and add 3 to it, the result is } of 20. Is vi 
more or less than 5 ? 
9 
, ; 3a, 
6. xis the cost of a house. A quarter and a half of it was —~’ 


—“—— 


which was the cost of two other buildings. The cost of all three was 
«+ 3a, or $1750. 
7. After x persons were seated in church 4 times as many more 
‘ame in, or ..a. When they went out, it was found that there were 
25 men and 3 times as many women. 

8. The long cars in a freight train are to the short cars as 2 to 3. 
There were 30 or 5x in all. The long cars are a. 

9. One of three brothers is a year old, another is 7, the third 1s 
half way between. w= 40f what? 


10. An iceberg moves xz miles a day, a ship 10 times as fast. 
When they approach each other, they are _. x x or 110 miles nearer 
in a day. When the berg follows the ship, they are _. x x miles 
further apart. 


9.— Analysis and I. Without solving, decide upon some 
Statement of quantity whose value is lacking, but may be 
Problems. found from what is given. Say— ‘let «= 


y) 


the distance from,” ete. 


II. Show what operations must be performed upon the known 


quantities to find the value of the unknown. Thus, 7 = ; 1 4 


1. It is 90 m. from New York to Philadelphia. Two men set out 
on bicycles to meet each other. One has gone 30 m., the other 55 m. 

2. Twenty rows; eight in a row; every other tree dug up and 
sold. 


3. 100 young apple trees; all but 25 bearing; 150 bu. the yield. 


22 APPENDIX IL. 


4. Two boys working together earn $1.50 a day; the first gets 4 
as much as the second. 

5. If you will bring 2 more like those I have, there will be enough 
POL OL: Us: 

6. I had to double my money 4 times before I could buy a $160 
horse. 

7. There were 400 grown persons at the fair yesterday, as shown 
by the receipts, which were $200. The same amount was received 
on children’s day for half tickets. 

8. There are 144 pegs ura shoe. costing 1 fa gross. At one bench 
24 gross are used in a day. 


10. — Problems If « = the sum of 5 and 6, or of any two 
Stated in General numbers, it can be found from the equation 
Terms. «=a+b,aand b representing the two num- 


bers, whatever they are. 


1. What kind of problem is to be stated as a = Fe x=a—b? 


2. What do 3 yd. cost at $0.80 a yard? Substitute a and b for 
the known quantities, and give the equation for all such problems. 

38. Burning a cords of one kind of wood, and } cords of another 
kind, what is the cost for m years ? 

4. If the cost for a year is d, the cost for each of n parts of a 
year would be what part of d? 

5. One person is } inches tall,-another '+ 1, a third b+ 2. 
Express briefly the sum of their heights as equal to y. 

6. Compare }+60+1+b+42; 3643; 3x (b41). [b=5.] 
If 3 x b= 15, why does 3 x (64+ 1) = 18, and3 xb}4+1=16? 


7. A pole x feet long is cut into 3 fractional parts, 
you take ; from 2, what remains of the pole ? 


8. A man paves a gutter in a days. What part of it does he 
pave in 1 day? In m days he paves m times as much, or YW. 


APPENDIX II. 29 


9. I walk how many miles in 0 hours if I go n miles an hour ? 
10. I go at the rate of how many miles an hour if in ¢ hours 


I walk d miles; i.e., in one hour I walk = miles. 


— 


11. — Terms. An expression that may be separated into 

two or more parts connected by + or — 

contains two or more terms. If it cannot be so separated, it consists 
of one term. 


os C ° . . : é 
As ab+5— 7 contains three terms, it is a trinomial. 
( 


es Ad wane ; 
As 5¢— a7 contains two terms, it 1s a binomial. 
v) 
a+b 
c 
An expression of more than three parts or terms so connected 
is a polynomial. 


As daxy and contain each one term, each is a monomial. 


Nore. — In the fraction a+b 


, the plus sign connects merely the terms of the numerator. The 


expression as a whole is a single term, a monomial. 

Remember that where there is a parenthesis marked by curves ( ), 
brackets [| ], or the vineulum , the included quantities are to be 
taken as one. 


Apply the names “monomial,” etc., to each of the following expres- 
sions, and explain the process indicated :— 


1. 3—(4—2) 4. 3(¢+d) q. (ae —20-— af 
2. a—b—c 5h. se+d 8. 3(a+4)—4(b—3 


38. (2—2x)—(y—4) 6. x(mn — 1) 9. (4a[b—2])(e+d) 


12.— Literal Powers 1. Define a power of any number. 2. What 


and does the eaponent show ? 
Numerical Coefficients 1st powers: 5, a, wy, (¢ — 4). 
as Factors. 2d powers, or squares : 


GF ax GO, a, oy, oxy xa xy, (C—9)’. 


30) APPENDIX II. 


Read, and give all the factors of — 
3. 0° 5. a°b T. ab? 9. (ad)? ll. (a+ 6)? 
4. wc Ged xc G. G0 10. a*b? 12. a*b®xy" 
13. How much is3 x5? 53? Explain the difference. 


> 


14. 3a=a+a-+a; the factors are3anda. 3a?=% 
The coefficient of a literal quantity consists of the factors pre- 
ceding it in the same term, usually the numerical factor, which 
comes first. Thus, 3 is the coefficient of 3 ayz, or 3 ay is coefficient 
of z. When no figure is written, 1 is understood: a=1 xa, or 1a. 


Give the numerical coefficient, and factor each term, thus: 


3m=mMm+-m+-m, or 3xXm; =m xX mMxXM. 


15. n 18. 2 a?m Bie 24. b*(a + b)— c(d — e’) 


a 

Oe ee eee LOS ry 22. 3ab—bB® =. Ba(b+c)+2y(f—g) 
9 

fs 300 20. 3 am? 23. 2cd+5z 26. ab(a—y)— 2) 


& 


Similar terms must contain the same powers of the same literal 
quantities. The numerical coefficients and the preceding signs may 
differ. 

mb and bm? are similar, but in each the 6 should be written 
before the m. 


27. Select similar terms among the following : — 


7 mb —2a*n 5 an — 2 vy Teal 
an 3 0'm? — 2 bm? 2 vy” — any 
— 4 oy? — 5 bm’ 4 ay? 2 bn —9an 


13.— The Value of 
the Unknown Quantity 
found by Subtraction. 


I. If the same quantity be taken 
from equal quantities, equal quantities 
remain. 


APPENDIX IL. 31 


Apply this axiom or self-evident truth in finding the value of # in 
these equations. 
1. «+4=10 — Subtracting 4 from each member, (1) « = 10—4 
Performing the process indicated, (2) « =6 


9. 4ea=3e24+17 


Taking 3a from each member, we have (1) 4%-—82=17 
Performing the subtraction, we find (29) Me AT 
8. x£+ 24 = 32 5. dl+a= 64 7. «+ $19 = $82 
4. B5&a@=—42+4+ 20 6. 27=15-+¢a@ 8. e+ $23 = $20 
9. $5 added to my money 10. The price of corn has risen 
will give me $28. How much 13¢ and is now 63¢. What was 
have I? it at first ? 
Let 2 = my money now. 11. The watch and chain cost 


Then «+5 = { my money after $5 $495. The chain cost $30. 
is added. 


Then x + 5 = 28, etc. 


12. Show that subtracting a quantity from each member of an 
equation is the same as transposing it to the other side with a minus 
sign. 


14.— The Value of Il. If the same quantity be added 
the Unknown Quantity to equal quantities, the sums will be 
found by Addition. equal. 


Apply this second axiom in finding the value of a. 


th e— {= 12 


Adding 7 to each member, we have (1)*%—-7+7=12+4+7 
Performing the processes indicated, (2)'%= 19 


What is the effect of taking away 7 and then adding 7 to any 
number ? 


2. a—$$12= $19 3. e—#=2 4. x — $24.75 = $ 82.75 


39 _ APPENDIX II. 


5. The cost was $18; the 6. He was in school 26 half 
discount was $43. What was days, and absent 12. How long 


the list price ? had school kept? 
Let # = the list price. 7. 327 miles had been trav- 
(1) « — $41 = the cost. elled. The journey was x miles, 
(2) « — $45 = $ 18, ete. and 487 yet remained. 


8. Show that in an equation hke «—3=10, adding 3 to each 
member is the same as transposing — 38 to the other side, with its 
sign changed. : 


15.— The Value of Til. If equal quantities be divided 
the Unknown Quantity by the same quantity, the quotients will 
found by Division. be equal. 


Apply this third axiom in finding the value of a. 


1 "0 Wee 24 

Dividing each member by 3, we have (1) = 5 

If 3a = 24, 1a or x = q of 24 or 8. 
Anel2ee 00 4. 2ie= 25 OU Sea 
8. 19y =57 §. 184 = $52.13 7. 2a=90 


8. I sold my house for $6,000. If this was three times the cost, 
what did I gain ? 
9. Seven times a number less 5 equals 51. 


10. Eight times my money and $40 is 12 times my money. 


16.— The Value of IV. If equal quantities be multi- 
the Unknown Quantity plied by the same quantity, the products 
found by Multiplication. will be equal. 


Apply this fourth axiom in finding the value of 2. if ao i 


Multiplying each member by 8, we have Cl) a= 21 
If 1 of x = 7, the whole of ¢ = 38 x 7 or 21. 


APPENDIX II. 83 


x 3 5a T2 
Wee a owe 4’ 90 6 KM = 8 210 
12 te 6 i? 


6. Lost 2 of my money, but $3805 still remained. How much 
had I? 


7. 35; of the whole distance and 21 miles cover the journey. 


8. 18 of the 30 miles of the yacht’s triangular course were sailed 
in 72 minutes. The full time was what ? 


9. A third and a fourth of my money made $18.60. How much 
had 1? 


(1) ; Z — $18.60. Multiply both members by the 1. c. m. of 3 and 4. 


10. 2 of what I received was silver, ? was gold, and the remaining 
$ 24 was in bank bills. 


17.— Reduction The value of # in any simple equation may 

of Equations. be found by applying one or more axioms; 
that is, by increasing, diminishing, multiply- 

ing, or dividing both members of the equation by the same quantity. 


The steps of the process should be taken in this order :— 


I. Combine similar terms; reduce to lowest terms. 
Il. Clear the equation of denominators by multiplying both mem- 
bers of it by their l. c. m. 
Ill. Transpose unknown quantities to the first member, and known 
quantities to the second, by addition or subtraction. 
IV. Combine similar terms. 
V. Divide both members of the equation by the coefficient of x. 


Pa Pere 
1. .Give 3 


4 
eh tae by 12, we have (1) 8a +9x2+ 108 = 10% + 192 
Subtracting 108 and 10a, (2) 8a#+9a— 10% = 192 — 108 
Combining similar terms, (3) 1@.= 84 
Dividing by 7, (4) G12 


+ 16, to find the value of a. 


2. Given “ = r+ Rather mots 6 = -— ah 17, to find the value of a. 


34 APPENDIX II. 


To verify an equation is to prove that its members are equal by 
substituting numbers for the letters that represent them. 


> M 

Thus, in the equation in Example pg: 3 mt ei —9= a + 16, we found that 
x=12. Using 12 in the equation wherever x occurs, we have 7 

5) 3 2 le ‘ 
eee Kes Rae SN Ty 
ss) 4 6 
24 36 60 

or 3 + — mi +9= gt 16, 
or acon iaettettr 
or 26 = 26. 


Find the value of x in the following equations, and verify each: 


3. 132 15 — aoe 8. oat ee a8 
PAZ a lost Se ie 9, 204 °2_17=0 
5. 108+ 20%4+14=127+4+19e 10. ange aa 
6. 18a—7416=52418 —2 1 Te 
7 e+et+e=_t 30 12. Pete a 5g 
14: 2a 2 451-983 
18. — Problems. 1. A horse and wagon cost $280. The 


horse cost 3 times as much as the wagon. 
2. Together the girls picked 45 quarts of berries. One picked 
4 as much as the other. 


3. Divide 23 into two parts, one 5 less than the other. 


APPENDIX II. 35 


4. The sum of two numbers is 87, and their difference is 9. What 
are the numbers ? 


5. I bought 6 lb. of coffee and 5 lb. of tea for $6.40. The tea 
cost twice as much as the coffee. Find the price of each per pound. 


6. Of three candidates at an election, A had 3 times as many 
votes as B, lacking 220, and C had 2? as many as B. If there were 
3300 votes cast, how many did each ‘candidate receive ? 


7. Four times a certain number divided by 3 added to 2 the 
number equals the difference between the number and 518. 


8. A certain number diminished by 40 is the same as 40 dimin- 
ished by 4 of the number. 


9. Take from a number 2 of itself, 4 of itself, and 40, and nothing 
remains. 


10. A man bought an equal number of horses, cows, and sheep. 
For each horse he paid $125; for each cow, $50; and for each 
sheep, $12. How many did he buy for $1,309? 


19. — Problems. 1. A’s capital in business is twice B’s. A 
loses $5,000, while B gains $3,000. They 
then have $15,000 together. What did they have originally ? 


2. A farmer sold 4 his wood at $3 a cord, 4 of it at $4, and + 
of it at $5. He received $44 for the lot. How many cords dia 
he sell ? 


3. A certain sum of money was divided among F, G, and H. F 
and H received $150; F and G, $216; and G and H, $178. How 
much did each receive ? 


4. Two tanks contain an equal quantity of water. But after 75 
gallons have been taken from one, and 50 gallons added to the other, 
one contains twice as much as the other. 


9. Divide 30 into two such parts that 4 times the greater shall 
equal 6 times the lesser. 


36 APPENDIX IL. 


6. Take twice a number from 19, divide the remainder by 3, and 
add the number. The sum will be the same as if half the sum of the 
number and 10 were taken. Find the number. 

7. The sum of three consecutive odd numbers is 39. What are 
they ? 

8. The sum of three consecutive multiples of 7 is 273. What 
are they ? 

9. A grocer sells 80 pounds of tea at 50 cents a pound. But this 
tea is a mixture of poor tea at 45 cents with a better quality worth 
65 cents. How many pounds of each kind in the mixture ? 

10. A has $250, and B has $75. How much must one give the 
other, that they each may have the same sum ? 


20.— Addition: the As in arithmetic only like numbers can be 
Signs + and —. combined in one sum, so in algebra only simi- 
lar terms can be added. To add da, 26, and 
leis in a sense like adding 8 acres + 2 bushels + 1 cent; we can 
only indicate the addition of the different units, thus: 3a+ 2b+1e. 
1. How many n’s are 4n, n, 2n, 13? 
2. Arrange similar terms in columns; then add — 
Zab, dc, Tax, 4bx, c, tab, bx, 3aa, Te, Lab, ax, bx. 
Indicate the sum of the four amounts just found. 


Suppose a man begins the week with no money, and in his book 
puts a + before each amount that he gains and a — before each 


Gain $ 3 amount that he loses. Instead of Lay 
Gain $14 entering each transaction as shown aay 
Loss $ 2 at the left, he might keep the avy A 
Gain $ 3 account as at the right. He would 4 34 
Loss $ 4 first add the + sums which in- ae 


crease his property, then the — 
sums which diminish it, and then + 19d and — 
put the two together. If it were CO etaed 
+ 19d and +7d that he put together, the sum would be + 26 d, but 
adding — 7d to + 19d has an opposite effect. The sum of his gains 
and losses is + 12d. 


$19 gain less $7 
loss = $12 gain 


APPENDIX II. 37 


Consider that the signs.+ and — belong to the quantities that 
follow them, showing their character as gains or losses, and we may 
write (+3d)+(+14d)+(—2d)+(+38d)+(—4d)=(+19d)+(—7 d) 
=(+12d). Adding —7 is like taking away + 7, and as all quan- 
tities are + unless marked —, we may write 19d —-7d=12d. 

3. Add a loss of $10 to a gain of $15; what results ? 

Add + and — quantities separately. 


4. —4a 6, + 227 6. 8 (b +c) —22 
+ 3a —Ilzl —2(b+¢) — 32 
+a 4+ 4 27] —3(b+.¢) nee 
—5a — 227] 5 (b + ¢) — 52 
+ 5a + 7 al (b + ¢) en 


21. — Problems. 1. A yacht goes 10 m off shore and 3 m back. 


Give these distances opposite signs, and add. 

2. Again she sails from port + 7m, — 6m, + 6m, +4m, — 10m. 
How far from port is she at the finish ? 

3. A man has no money, but there are due to him $10, $3, $4.50, 
$2.50. He owes $2, $6, $0.75, and $1.25. If all these sums 
should be paid, would you mark the balance + or —? In adding 
the eight sums given, which four should be marked — ? 


4. A bin is kept for corn and oats mixed. One bagful = (¢ + 0). 
The changes in a month are 10 (e+0) received, 4(c+0) sold, 25(c+o) 
received, 2(¢ + 0) received, 9(¢ +0) sold, 1(e+ 0) sold. Arrange 
in one column with the proper signs, and add. 

5. Suppose a man’s debts are $100 greater than the money that 
he has together with what is due to him. Which will represent the 
amount of his property, + $100 or — $100? : ¥ 


6. If the sum of the — or negative terms is greater than the sum 
of the + or positive terms, will the result be + or — ? 
7. If you go forward 100 miles (+ 100m) and backward 150 miles 


(— 150m), is your final position in reference to the starting point 
+ or —? 


38 i 


Negative Quantities. 


When plus means 


APPENDIX II. 


Any quantity may be taken in two opposite 
senses, positive and negative, which we indi- 
cate by the signs + and —. 

1. What is the difference between 10° 


22.— Positive and 


sty ‘negative above zero and 5° below zero ? 
above = x | below 2. If we take away a man’s gains (+ 
forward : _ back quantities ), is his property diminished or 
} before Eater increased? If we lessen the amount of 
: ae, é inca his losses (— quantities), will he have more 
excess \ deficiency dollars or fewer? Taking away a posi- 


tive quantity makes the minuend smaller. 
Taking away nothing leaves the minuend the same. 
Taking away a negative quantity makes the minuend larger. 


10 —0= 10 


10 —(+3)=7 


Let us compare these four cases : — 


To add 
——_sSa—_— eee 7 


Like signs. 


+ 6 — 6 
ee Rae 
Pine 


Here the first 
quantity 1s in- 
creased in the 
same direction 
by adding the 
second quan- 
tity. 


Unlike signs. 


30 eb 
pee eae 
42 —2 


Here we com-| 


bine opposing 
quantities. <A 
part of the larger 
is counteracted 
by the smaller. 
If the negative 
is larger, the re- 
sult is —. 


10 —(— 8) =13 - 


To subtract 
a REET a BN EME RR RET a 


Like signs. 
+6 —6 
+4 —4 
A) ae 
Here each upper 
quantity is les- 


sened by taking 


away 4 of the 
same kind, leay- 
2 of each kind. 


Unlike signs. 

+ 6 — 6 

an ta 

+10 —10 
If taking (+ 4) 
from (+6) leaves 
the (+6) smaller, 
(—4) from (+6) 
leaves it larger. 
If (— 6) denotes 
a deficiency, tak- 
ing away (+4) 
will increase the 
deficiency to 
(— 10). 


APPENDIX II. 39 


23. — Subtraction. 1ee2-from d= * LO. from 0 =. 
2. —2and —1? —2 from —1? 
Subtracting a positive quantity is adding a negative, 
and subtracting a negative quantity 1s adding a positive. 
38. From 5 7 9 —3 
subtract ys 0 —2 1 To subtract a quantity, 
Abie 9) 11 1) change its sign and add. 
subtract —6 —8 8 8 Ee, 


5. Read each of the preceding quantities with its sign, thus: 
+5 less + 2. 
In such operations do not confuse the sign of subtraction with the 
sign which marks a quantity as negative. 
2—1 may equal (+ 2)—(+1) or (+ 2) +(— 1). 
What number must be added to the subtrahend to produce the minu- 
end in the following cases ? 
6. From a —a b—a Ve2 Geo Cx Hay 
subtract b b—-—a—b a —3b —6a —10ay 
8. If a= 5 and ) = 3, compare in value —a+) and b —a. 


By changing signs and adding, — 
9. Subtract 19 —8 —T 4 10. w« 3y —a 42 


from 17 —6 4-7 22 2y —x% —8z 

inh 12. 13. 14. 
Min. 2a+sb a—3sd 4m+5n—- 2 In 18 is 5n a part 
Subt. ime -2i—. 2 OM —44 of what remains ? 


Shorten the expressions : 
24. — Terms in l.at+tbd+e—bd 4. 2% —S8ma—a+ mer 
see Da 2a+2b—c+sa 05. 8act+z—ac—z 
LE are Sac 38. W—2ab—a@ 6. 4a? —2y?-— 3a*+y 
7. Explain the difference in value between 
6— (842) and6—342 


40 APPENDIX IL. 


If +3-+2 is to be taken from 6, we change both signs and write 
6—3—2. If the quantities in curves were to be added to 6, there 
would be no change of sign, for 6+ 3—2=6+4 (3 — 2). 

Remove the curves, etc., without changing the value :— 

8. T—(8+8); 44+7—(4—2) 9. a—m—(b+ce); a+(n—c)—(m+e) 

Remember that an expression in parenthesis is a single term; 
thus, 8 —[4— (2+ 1)] becomes 
first 8 —4+(2+1), (2+1) hav- 
ing its sign changed as a single On removing parentheses 
ore after a — sign, change the sign 

Remove parentheses, making | Of each term they enclosed. 
change for one pair at a time. 
Combine similar terms in the result. 


10.9—(8—3) 12.9—[4+(2—1)] 14 2ax—(8az—[y—aza)) 
112¢ — ow 13.¢7¢—[a—(#—y)] 15. (2be+n) —(n+2 be) 


Without changing values, enclose in parentheses the second and third 
terms of :— 


1I625 2 See? lisa 3-3 —¢--2 18. «—-y+2z+a 
25.— Multiplication: 2 x 3 ft. = 6 ft. 26 Sto ee 
Product of Coefficients; 2x 3a=6a 4ab x 127° = 48 aby’ 

Sum of Exponents. 20% 30 60D Tm x 9n? = 63 mn? 


In multiplying algebraic quantities, the product of the coefficients is 
taken first. 


1. 5a x 2be 3. Da x 4ay 5. 19 ax x 12 by 
2. 6abx 3-2 4. 12bx4cy 6. 3h be x 18 dz 
7. What does an exponent show ? 

oe 0 adie Uae bp 30°b x 3.ab?=9 a0? 
axaxa=o 2 Ki ae Hide ral 
GXOX DROS = Xo ee 3m x 2m? = 6m 


8. Show how the exponents of each product (above) were obtained. 


APPENDIX II. 41 


In multiplying algebraic quantities the exponent of a power is 
found by adding the exponents of its factors. 


Notre. — As in arithmetic, so in algebra, the order of the factors does not affect the product. The 
alphabetical order is generally followed. 


9. Leta=2, b=8, c=4, and show that abe = cha = bac = cab. 
Shorten and rearrange the expressions : — 
10. warcb 2 cha ll. aamec*na? 4. 
Put all the factors of each group into one shorter tern : — 
12. 2ab x 4ab 15. (m + n) (m+ n) 18. 6 (a — b) xX (a—d) 
13. 3be+ avy + 2 cw 16. a’a x aa? x bac 19. (m + n) (m + n)? 


14. a}, a’, at 17. 3(a—b) x 2 20. 3(@+y)x 4(~@+y)? 


Explain : — 
aioe oa + 2 4 oF et+ty+z2 23. T7ab+3bc+4cd 
5 a 2 ac 
15 — 10 + 20 ax + ay + az 14 abc + 6 bc? + 8 acd 
Multiply : — 
24. w+yPte 25. P+y+2 26. (n—n)+m+n 
Bey LY 36 


27. 2ary+ day? +4 xyz 
5 aay 


28. In finding the product of 2a+36+ 6c and 3+4, how many 
times is the multiplicand to be repeated? Find the two partial 
products and add them. 

Similar terms in the partial products must be written under one 
another and combined in the result. 


Find and add the partial products : — 


29. a? +4ab+0? 30. 24+ 38ay+4bz 
a+b 2az+ 2 by 

a’ + 4.a7b ete. 

+ a’b etc. 


81. F& +2m+y+m 


etmt+ 7? 


42 APPENDIX IL. 


26. — Multiplication: If 3x (+5)=+15, 3x (—5)= —15. 
Like Signs give + ; Un- If 3xa=a+a+a, 3x (—a)= —a—a—a, 
like, —. or —3a. 
If multiplying z by 5 means adding & 
repeated 3 times, then multiplying « by — 38 means subtracting x 3 
times, or —®—a@—aw=— da. 
When only one of the factors is negative, the product is negative. 


Read the products :— 


de 7 —4 —ab 2 xy ab — ay? 3 ab? 
—7 12 3 —4 ay —4a*x 


SS eee 


2a—b 8 2¢0+38b 4 38an—4m?+mn 5. 2ey+y+ab 
30 —3 m+1 x—y 


If we indicate the product of 4 multiplied by —2 as 4 to be 
subtracted twice, or — 8, we may write the product of — 4 multiphed 


by — 2 as 4 to be added twice, or +8. —4 times —-2=+4x+2. 
6. —sab*—2Zarx T. 4mn?—4n*+7 8. my—38y—n 
— We —2mn — ay 


9. —(? —3 ay — 9) x (— 2+ ay —y) = 


27. — Division: 1. Division is the inverse or opposite of 

Quotient of what process ? 
Coefiicients, Difference “9° In 3a x 4a° = 12° (show howewoert 
Ge redhat cE cient and exponent in the product are ob- 


tained? 38. What terms in division and 
multiplication correspond to each other ? 


12a Pe ei 
—~—=44'; es eS 
30 4 a 
5. Show how the coefficients and the exponents of the quotients 


were obtained. 


» &o X4aaaaa é 
2, — oo = 4 a, 
3 ad 


4. Explain: 


APPENDIX II. 43 


Give all the factors of the dividend, and show which remain after 
tuking out those found in the divisor:—  ° 


a 5 ache 8 a7? 7 yee 4 
6. 3 xy 7. 15 ab? g Saye 9. 4(a + b)(a — b) 10, @. 
x 2 ab 22 a—b 0° 


Divide by giving the quotients of coefficients and the difference of 
exponents of the same letters : — 


iL 12 are 13. sah ans 15. iB nae = bs ard 17. 13 a(@ + y) 
6 ab? 8 b°x 4 a? 13(@ + y) 


12. 18 atm’? 14 14 ara 16 20 mt + 24 m? 


aC ra 7 18. 900 +453 
9 am 2 ax 4 m? 


19. What is the law of signs in multiplication? 20. When the 
product (dividend) is negative, what signs have the factors (divisor 
and quotient)? 21. Show that the law of signs in division is the 
same as in multiplication, — 


Like signs in division give +; unlike signs, —. 


22. The dividend is 6. Will divisor and quotient have like signs 
or unlike? 28. Dividend is 6; divisor, —35. Why should quotient 
be — 2 rather than + 2? 


24. Dividend is — 2a, divisor 2, quotient ? 


First give sign of quotient ; Give product of quotient and 
then its value : — divisor : — 
Drvisor. DrvIpEnp. : 
cD re a Les ay Cae en San amy oY a ae 
25. a+b —(a +d) 30. 2 vy)6 wy eae £2 xy 
26. —ab — ab‘ sa—Ly + 1 
27. — mun 3 mn 81. —38y’z) 38 y2—9 2? +12 yz 


28. —6(x—y) — 18(x — y)? Macy oe 
29. —T(a+2)? —21(a+2)° 


82. ax)4 wla—3 ato? +2 atx? 38. —8 x°)—8 b*—8 by +16 y! 
84. 6 b’y)6 b%y — 12 bby’ — 12 bey? 


44 APPENDIX II. 


28. — Factoring 1. What quantity will divide every term 

at Sight. of the trinomial aw+ba+cx? What is the 
remaining factor ? 
2. Find the numerical and literal factors of 3am +6an—9 ay. 
How is the third factor found ? 
In the same way find two or more factors of — 

8. 6a—3b’y 4. 5at*—6a 5. Bax —12 a’y + 24 ay" 

6. Find the two prime factors of 4a+ 8ab —12h 

7. Of 3a°y — ay’? + 2 ay 

8. a, b?, and 12 are factors of what monomial ? 


Give two factors : one a trinomial, one the largest possible monomial. 


9. 200— 6070 +4007 ll 9abh 450275 
“102 3 ey + 12 ay? — 1b 377 12. —12 mv —16 my — 6 may 
13. What quantity will produce 2a when multiplied by 1? By 
—1? 14 1isa divisor of 2a+6; what is the quotient? 15. If 
—1 is one factor of —a—b, what is the other? 16. Give the 
square of 2; of — 2. 


29. — Fractions to All operations with fractions in algebra are 
Smallest Terms. based upon the same principles that apply in 
arithmetic. 


Reference may be made if need be to their development on 


pp. 54-72. 
3 ei} av a 3m’y 
1. Explain: —— = — —— = — : 
caer 2 Db ee 18 6my 


To change a fraction to smallest terms, — 


—m 
2 


Strike out all factors common to numerator and denominator : — 


A gy? 5 12 atd’e 3 28 mna” 

 Qaryp ' 150%? © 85 mna 
3. A atx 6. 27 aPary? 9. 21a(a— 6°) 
16 aa 36 ax’y 33 b (a — b*) 


4 ab—eb 7. 24 a’bx* 10. 51 abed (a — 3) 
abe 30 abm? 68 a*a (a? — 3) 


APPENDIX IT. 45 


30. — Fractions to 1. What is the effect of multiplying both 
a Common Denomi- terms of a fraction by the same quantity ? 
ator. ~ , 2 
grates 2. Change i and 5" , 0 a common denom- 
c “¢ 


ac 


inator. Explain the three steps of the process as shown below. 


if nF LE 
Sel Oe d 4bed _ g a xd Pa dx 
4 be 4bexd 4 bed 
4be=4xbxe Abed _ 5, Ree 2005 2by 
Qed 2cdx2b 4bcd 


4be x d=4hbed = 1. e. m. 


To change fractions to a common denominator : — 
I. Find a common multiple of the denominators. 
Il. Divide this multiple by the denominator of each fraction. 
III. Multiply both terms of each fraction by the quotient of its denomi- 
nator into the common multiple. 


Change to common denominators : — 


sy @ 1 am x y Zz 
| A I CRT, : Ae : alan pos 2) bag 
2ab’ 10¢ Qey 32° sabm* 6b?m ad? 
31. — Fractions To add or subtract fractions : — 
Added and Sub- I. Change to a common denominator. 
tracted. Il. Place the sum or the difference of numer- 


ators over the common denominator. 
Ill. Change results to lowest terms. 


4a, 20° 2 Ye) aC m 
1, ——+-=— A i 7. =---—— 
ty xy Buy wy z b dn 
9 &, a, 4 5 1 2 ad gee Nw 
By oa og ey ey ced’y? dy? 


9 9 Y b? 


g, 2@_3o g, 34@—4) | 8a+4) 
i Y 


AG APPENDIX II. 


ee Ae PRE 2 4 
32. Ree ea 1. Explain: Be Re ae = oa 
Mim bic 


of Fractions. be B? 
To multiply fractions together, — 


Write the product of the numerators over that of the denominators. 
Cancel.when possible. 


Note. — Integral quantities may be given a fractional form by writing 1 as a denominator. 


9 tan, 2b de 5. ae+yY) y 2 abe? 
dby av 4 b°cd* b’ (a@ + y) 
3 3a'b 4y Onn 6 (a—b+o)x l6aby 
167 6a'b Seen ax — ba + cx 
4 9 mna . 50 dtay? 7. (a + b)? 2 (@ + y)* 
25 Bary 27 mnz (7o+y) atb 
Sib cee. el 1. Explain: OW 8) 
Fractions. 0 bys Dye 
or per ae ie ag rt Loa aa 
DV AOL ee 


In division of fractions, — 
Multiply the dividend by the divisor inverted. Cancel when possible. 


Nore. — Integers may be given a fractional form as in multiplication. 


9 2em , mM 5 ab |. =e BL oes 3 aba? 
. ay . y , x 2 . Y of 
2; a eke ste) ~ , 272 
3. 2dz__10dm F Mia pe ola ie) 9 (%&y& ode 
5m 5z Oey: 6 ay” y dj ytd 
4 a+b, (a+b)e 7 abed , 2 ab? 10 —2ad* _ —4ard 
Tice ie y age Tay 460 Abe 
34. — Equations with (1) e+y=12 (2) «—y=2 
Two Unknown Quan- These equations are independent; that is, 
tities. neither one is made from the other. They 


are also simultaneous; that is, the same 
unknown quantities have the same values in each equation. 


APPENDIX II. 47 


nel) see 24 Are these equations independent? (2) 


ay we - is made by doubling (1). Are they simul- 
(@) 2a+2y=24) taneous? 


Are these equations simultaneous? Can 
((1) e+y=12) 4 


2. fe G f the sum of the same two numbers be both 
(@ety=1T) 49 ana17? 


8. Let e=3 and y= 2, and make two equations that are both 
independent and simultaneous. 


(1) 2e+y= 13 If we combine two or more independent 
"(Q)8a—-y= 7 simultaneous equations in such a way as to 
—————~  eause one of the unknown quantities to dis- 
Pane aida appear, we eliminate it, and obtain an equa- 
Whence @= 4 tion with one unknown quantity only, which 

and 2%=8 is readily solved. 
(1) §8+y=15 In Example 4 we have added the equations. As + y 
y= and — y neutralize each other, we obtain equation (3), 


5*%—20and%=—4. Substituting 8, the value of 22, 
for 2a in equation €1), we obtain 5, the value of y. 


5. Verify the equations in the preceding example by substituting 
their values for x and y. ; 
(1) a +-7 y= 31 In Example 6 we eliminate « by subtracting (2) 
: ; from (1), and find values as before. 
(2)a—3y= 1 
(3) 10¥=80 
y= 


It will be observed that we eliminate by 
addition when the quantities to be eliminated 
have unlike signs, and by subtraction when 
a+21=31 they have like signs. 


(4z—y= 9 (Txe+y=99 
( aty=16  (9a+y=T5 


9. The sum of two numbers is 19; their difference is 11. Find 
the numbers. 


_— 


10. Three times Charles’s money plus Henry’s = $325; but if 
Henry’s be taken from four times Charles’s, $ 200 remains. 


48 


35. — Elimination 
by Addition or Sub- 
traction. 


Process. 

(1) 4%—2Zy= 28 
(ZY da+3y= dT 
(3) 12e@—6y= 84 
(4) 10%¢+6y=114 
(5) 22% 195 
Cie i 

(2) 454+ 3y=57 
oye 12 

y=4 

(ety =13 
(38a—2y=14 

ay OS eae 


(8a+4y=61 
wh (2e+y =26 
- §da—3y=27 
(3a —2y=16 


APPENDIX IT.’ 


Find the values of # and y in 
\ x—2y=28 
oe+3y= 57 
I. We choose to eliminate y because its 
coefficients are the smaller. 


II. We make the coefficients of y alike by 
multiplying equation (1) by 38, and equation 
(2) by 2. 

III. We eliminate by addition because the 
signs of the y-terms differ. 


IV. We find the value of wx by division, 
and of y by substituting the value of 5a 


in (2). 


F (6x%—10y= 24 10, S*tzy =18 
2¢— 44.296 llety = 8 
7 (8e—5y = 11 4, fee—-Zy= 0 
 (10a%+ 6y =150 (3a —2y = 25 
9 (4a—2y = 28 19. (3e=43+4+y 
 (8a+5y = 47 (Ta=6749y 
9. (5a—4y = 7 13. (dy—4z= 33 
(7e+3y = 70 l6y+5z2 =118 


Norte, — First clear of fractions after reducing them to lowest terms. 


ey 
ba 
14. 
ne TD 
SING 


ory i 


15. 


OFF Or 

Or 
2 

a 

|S 

| 

bo 

— 


Lies 


) 
16. < 
oe eg 
[ ia 
i ae 
17. 


) Sa 4+27 18 “ba tay 
— os A 91 
6 Soames ¢ 


APPENDIX II. 4Y 


Bea+4 ,5y+3 op ae a ey 
eit = 29 a ai J 
_ aE: eae aha a faa 
pa 2 oy —S lg Ty 22 by 
al b ai Chad eet eee 
| Seer | gstyt*-—3 7% 
36. — Equations 1. Three times A’s money added to 4 times 


with Two Unknown b6’s money is $310; but 4 times A’s put with 
Quantities. Problems. 3 times B’s makes $320. How much has each ? 


2. A and B together have $850. <A spends 
$150, and B earns $100, and then they have equal amounts. How 
much had each originally ? 


8. Divide 100 into two such parts that 3 of the greater shall be 
equal to 2 of the smaller part. 


4. Half the sum of two numbers is 32, and 5 times their differ- 
ence is 80. What are the numbers ? 


5. Six pounds of tea and 4 pounds of coffee cost $6.10, and 3 
pounds of tea and 8 pounds of coffee cost $5.45. Required the 
price of each. 


6. Two men together own 100°acres. If A sells 1 of his share, 
and B gives away + of his, they will then own 55 acres. How many 
acres had each at first ? 


7. 30 is the quotient when the sum of two numbers is divided 
by 3, and 4 of their difference is 4. What are the numbers ? 


€ 


8. Of two numbers, 3 times the larger less 56 equals 4 the 
smaller, and 2: the larger added to 4+ the smaller equals 11. Find 
the numbers. 

9. 2 of .A’s property is equal to 3 of B’s, and the difference 
between A’s and B’s is $18. What has each man ? 

10. Should 3 be added to the denominator of a certain fraction, 
it would become 4; but should the numerator be increased by 2, 


ise 
it would become 3%. What is the value of the fraction ? 


FS ee 
SSSSSSSSSeASAR SHES 


SO 90SEC. Ole hs 9" ODES 


ANSWERS. 


Art. 23. 
. $45.40. 


500.49. 
3915.24. 


35751.37. 
48590.80. 
11465.13. 
15729.55. 
15850.62. 
16197.45. 
14681.69. 
16678.86. 


Art. 34. 
$ 16654.13. 


4881.11. 
4194.20. 
7198.54. 


12726.65. 
15138.25, 


6668.01. 
3960.21. 
9668.11. 


14113.23. 
17595.28. 
25258.24. 
21218.38. 
16970.48. 
16985.86. 
15912.48. 


seems CF oh: he Ce: RO: ee 


_ 
° 


Al 


Art. 24. 


$ 6410.05. 


6740.79. 
6615.78. 
6591.27. 
5988.19. 
7614.95. 
8611.69. 
9851.75. 
6641.52. 
8302.91. 
9628.47. 


10524. 15. 
12211.41. 
18258.12. 
14691.68. 


Art. 36. 


$ 473.23. 


$ 30447.10. 


$ 140.53. 
509.52. 


$ 127.738 loss. 


256506. 
860758. 
850 feet. 
226 d. 

$ 491.61. 


oR 


Ver 
$ 6378.14. 
6712.69. 
6419.09. 
5764.79. 
5267.56. 
6598.78. 
8106.48. 
9539.25, 
6164.75. 
6875.45. 
8700.77. 
10266.72. 
11669, 24. 
12921.62. 
14200.07. 


Art. 38. 


. $10.18. 


500.78. 
8458.25. 
15999.60. 
7156.52. 


Art. 43. 
4758, 
6516, 

$ 1.75. 


OM UAS 


_ 
S 


—_ — 
SOMAAMP WN SCOMRIAAR WYP 


OOARD AP w 


$ 1548. 
$180. 
$ 240. 
$6.21. 
50h. 
$ 104. 


Art. 45. 


28578. 
59448, 


. 61101 Ib. 


41909. 
6,797,200. 
$ 96312. 
258,138,200. 
. 60,195,000. 


. 62000 in. 
. 8,060,000,000. 


Art. 48. 
. 453,456. 
2,373,672. 
15984¢. 
35441. 
114,163. 
$ 3,019,365. 
62464 oz. 
69223. 
7,928, 712. 
$ 38096. 


Art. 49. 
$ 5748. 

$ 131,486. 
$ 128,945. 
$ 12592. 
5551¢. 
$55.51, 
137425¢ or 
$ 1874.25. 

. 198186¢ or 
$ 1981.86. 


OE 00S Clg St aR Oe oa 


_ 
° 


Sot Or CAE an CO5 NF 


ANSWERS. 


11. 1,170,000,000¢ 
or $ 11,700,000. 

12. 13500¢ or 
$135. 


Art. 52. 


. $316.80. 


2010. 
65.16. 
4862.97. 
1850. 
717.60. 
9372.80. 
6420. 


ATCO: 


528. 
1237. 
275. 
310. 
653. 
57. 
132. 
1501. 
26. 
7200. 


Art. 59. 
7142, 

1900. 

75 yds. 

70838. 

800 bales. 

25. 

197 or $461. 
200 times ; 22 
will remain. 


. $1608, 
| STH. 


Sr Set Se tues noe ene es 
OP" So Oa etree 
i i a 

et col oe ne 


ow 


TSnHes Aooes 
ao Ww by is ba 
Se et rt | RR 
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RR RY Rie 
colo ol” col 


PPO Pp 


| 


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. . 

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| 

ol 


Soe See eS SSS OSS 
aoanNrtanraedaIQ on Pp 
Ol e'CO colon COIN C[AT SOUP colo OP Olt cojn 
OJOS cole colo coloo CO[LT cols colts colon col 


sana eee ee 


. 62. 


| 
By 


HR | 
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4) 
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oy 


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sles Lay 
apo oy 


aaoanrFrwWhN WY Re 
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| 
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BRR Ry ey, 
ao 
i? aj a) ~3} 


Oonr PR OD DE 
wie bo bie RR +, H| 
Nap Bl ilies alm al? a a7 alto a 
Rona Fw w WY eR Re 
Ri eRe lho i lee 
so ae 


tdi bey, 
rw | 


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“leo bole be} 


bol by 
Slo o} 


IAM Pp WP BRR oe wD Oe 


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a. 


BO bolt BIO bol BIO DO BIO Bie 


RrPwwnwm Nee 


Loe bd} 


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Do + LO), bolt boILO bot boItO Bl rol Wl pty, 


Nr Dp 
Rio | 
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Ov CO Pe B® PRP OOD DY ee 


1 Oe Co 
aR 


wen clo cjo 
° : ° 


772 ; 294 rem. 
b. 1794; 
c. 13825; 507 rem. 


“et See Foe 


Sag Roe) OF Fe 08 eer 


_ 


829 ; 202 rem. 
414; 775 rem. 
485; 854 rem. 
1395 ; 246 rem. 
734; 618 rem. 
402; 501 rem. 
371; 644 rem. 


joo to 0000 GIO 
A . ° ° 


OO HOO HG OKO pl 
OO HO WIP DM ° 


CON FO RO RISO LE 


ce] 
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Cpr C709 Ores HA} 
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sale i 


Art. €5. 
$ 921.42, 
$ 144.99, 
1254.35. 
$ 478.94. 
$ 1632.59. 
$ 53.66. 

$ 2165.19. 
$ 230.55. 
5346. 

$ 225.63. 


ANSWERS. 


Art. 67. 

565 1b. 

$350. 

$ 525. 

$ 0.75. 

28 m. 

$2500., each 
man’s gain; 
$80000., cost of 
land; $95000., 
proceeds of sale; 
$15000., whole 


Dap ww 


ae 


gain. 


8. $15.75. 

9. 3¢ cheaper to 
send by mail. 

10. 22 qt. 


Art. 68. 
8. 2640. 
9. 17741. 


10. 15924. 
11. 15522. 
12. 1,206,060. 
13. 32616. 
14. 3690. 
15. 11466 ft. 
16. 10710 lb. 


Art. 72: 
rR 128.08, 
y = 148.92, 
Zz = 209.39. 
Sum of A= 
1180.11. 
Sum of B = 
693.72. 


Sum of A— sum 


of B = sum of C 


or 486.39. 
8. 1,553,424249 
4. $1225. 


See aT e OO ves Or ae 


SOs ee Coe ap Nene, bag 


=" 
> 


Beano eat Si CA «pms 009 


- 
- © 


1652 Ib. 


. $109.85. 
. $1050. 
. $1.50. 


Hans. 


$4. 


Art. 77. 


$16. 
$ 192. 


- 4608 in. 


84480 ft. 
5185, 

50 d. 

B41 ee: 
245. 
128, 


Art. 83. 


$ 1632. 
152.38. 
210.11. 
655.13. 


1107.76. 


533.80. 
21.46. 
25.66. 


LIVELT: 


39.36. 


inn te hey 
$ 1097.10. 


$ 124.20. 
13255 Yr. 
$8100. 
1154289, 
27200. 
110. 

6288 lb. 
9504 lb. 


504 Ib. 


cin hoa eee 


163 


_ 
S 


Ce eerie 


OM AUD 


pepe hel ot. ta, ke 


Art. 88. 
83,8; d. 
1750 rm. 
3122 bu. 
694 54, sq. ft. 
390 8% ¢c. 
5625 sq. m. 
115428° cu. ft. 
282169 m 
2894 boxes. 
$2 T. 


Art. 89. 


26000 pckg’s. 
18944 pt. 

5824 pt. 

$ 5,366,760. 
2500 quires or 
60000 sheets. 
86400. 

443520. 

12375 ft. 

4840 sq. yd., 


43560 sq. ft., 
6,272,640 sq. in. 
. 8,317,760 cu. in. 


Art. 90. 


$ 38.40. 

$ 11424.89, 
$6198.43. 

$ 5255.57, 

$ 604, 

$ 51.60. 

$15.08 saved 


pe buying of 
second firm. 
. $17280. 


$ 19430.65. 


. $22076.28. 
. 145.8 ft. 


13. 
14. 


. 7 passengers 


fewer than 
when $110.55 
was received 
from 33 trips. 
$ 0.02. 

8335) in. or 
6944 ft. 


Art. 96. 
1. 51,973,650 ° 
coins. 
2. $35,506,987.50 
3. $14,989,278.60 
4. 2640 ft. ; 
o20 ft. ; 
660 ft. ; 
4620 ft. 
§. 4320 sq. rd.; 
5200 sq. rd. 
6. 23528 cu in.; 
315 ft. 2 in. 
7%. 253,800. 
8. 6696; 58590; 
334,800. 
9. 3180 lb. iron 


10. 
11. 


for 1 ]b.silver; 
716 lb. iron for 
1 lb. nickel ; 
8 lb. iron for 1 
lb. lead. 

37 854, cu. ft. 
$ 91.25. 


30 - 

+ bb]. =24¢4 lb.; 
$0.35 gain on 
1 bbl. 

$100 = 

£ 205940 ; 


10. 


— 
oO 


> 


OOO Oe OE Na et 


ANSWERS. 


$100 = 
518,28, fr. ; 
420 40, M. 


. $0.674. 
. $2 13 ; 


52 wk. 2d: 


. 49 gal.—11319 


Cleanse 
3 bu.=6451,26 
cu. in. 


. No profit. 
. 332° bu; 


$ 19.8331, 
100 bbl. 
63 bbl. 


Art. 101. 
24. 

420. 

ibe 

2112 mM, 


. 42 periods. 


$ 9216. 
13m. 
$ 270. 


. 30 pieces. 
. 4.53 8-hour 


watches. 


Art. 102. 


. 388492 deposi- 


tors and a re- 
mainder of 
$ 181.04. 


. $1075.57 ; 


$ 185.43. 


. 888,24, ft. 
. $610 loss. 
. Sun’s diame- 


ter = 1093338 


— 
i 


Soe Rs Oak uene Co. tot 


x earth’s di- 


ameter. 


. 234 min. 

. 4164 lb. 

. 2,280,960. 

. 92,795,826 m. 
769.2 -— 1m, 


oF 
on 
no & 


oe) 


BS 
Ne) ~! DS WlR wo 
Ss CO Co see 
: his 
oo 
ge 
~ 
— 


HA\bo 
|co 


Rie 
00 eo 


oly wis +] 
so 


Oofoo bs] 
© el et cs of 
° ° ° ; 


to 
wi? 


11. 
12. 


eee Ol ie, OE 


Loe 


— 


OojUT CYDO BOHN cole * 
. ° ° . a 


Col Ol * 
m . 


mS one RR RS PO Oe ae RS eS On 


vo ° 
bon 
> 


bo} 
sepa 

1] 

° 


et | 
Ri oO} 
o, 


2 ah ye 


. 288. 
. 1020. 
. 12000. 


Art. 119. 


— 
COR. 


ee ee oe 
aS OV 


_ 


_ 
S00 Ott Oe Se 


873 ; 

6 ro00: 
$235 

403 doz ; 
27 oy cd. 


— et ore it 
Hal Ol Co Rl Oco 
on ° 


~I 


* ow oo ° 


»| 
| 


on 


we bd aT bw eH + 
08 Ts a J] vie 
ne a On Mh ale ° 

. . on 


~I 


31 
is 455 great- 
ie] 
than 3’5. 


oO 
Fr ol 


. 123. 


- 


OMIA TP w 


. 84, 


SOMATA WD 


CHOIR MAP WW 


ANSWERS. 


Co 


ON RR RR 
i eis Rl 


ni 
mor Oo 
e ° 


7 OK} 


pits ST et on 
co 
or . 


a 


20% %. 


. ify TE 


173 
. 8550: 


Art. 124. 

the un- 
known num- 
ber is larger. 


er, oo 
oe 


mi2o. 


Art. 129: 
$ 1 Ps: 
8331 sec. 
6560. 


. 51063. 


$ 71.038. 


. $160,781. 
, 681% Ib. 
. 2814 Ib. 
. $173,958. 


(10. 38975 m. 
11. $2013. 
12. 5534,%, m. 
13. $2943. 
Art. 130. 
1. 6 yd. ; $0.513. 
2. $2.69. 
8. $149.50. 
4. 4943 ft. 
5. 80,; cd. 
6. 102 sec. 
7. 45) m. 
8. $38.99. 
9. $0.09. 
10. 1161, bbl. 
Art. 133. 
1. #43 
2200. 
3. 56}. 
5. 193. 
Gre 0. 
yiiec e 
8. 84. 
9. $$. 
10. 1222 
Art. LSS: 
1. $2.914. 
9. $2.952. 
8. $0.51,%. 
2.50. 
4. $0.79}3 
0.29 Ps. 
5. $0.50. 
1.83} 
6. $933 
7% $3.94 
8. $45.80 
9. $2.85. 
10. $71.14 


_ 
~ 


— 
sO 65.00 =t Scr 1 Cop 


a" 
FS Ot Sa. SR GeO” 


. $3.62}. 
12. 
13. 


i, 

$ 0.57%. 

$1.60 is 20% a 
gross better. 


4 


. 140. 


ko 
2 


_ wl 


ou 


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. 


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| 
ow 


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| 
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Wir CIT Ol 
loo ° de 

; “| 


pt et 


olen ON 


ol olor 


| 
ol” 


bo 4 

oR 
aw | 
° ° 


11. 11,333, Ib. 
12. 161 ft. 


Art. 143. 


. 581 times. 
6,47, sq. rd. 


OTA N 


: EE eoocnS. 
$ 0.411 re- 
maining. 

9. 1075 m. 


Art. 144. 


Cf le 


WIL, RIS OlOe Cojo bo} Col 
_ 

o ow? ‘eg 

° ° 


i 
/ 
oO! ad 
S| 

° 


pa Toh Of 
Woe 


no GR NID RR KR | 
FV ell Da 
° 


106 ; 50 rem. 


ANSWERS. 


(9) io: (10) 2433. 
(10) 13335: 8. (1) 22 2. 
4. (1) 15 (2) 46°; 
(2) 1y5 (5) 8037 
(3) 134: (4) 7455- 
(4) 153: (5) 101; 
(5) 34. (6) 112}. 
(6) 13% (7) 263. 
(7) 135 (8) 843}. 
(8) 1 (9) 5523. 
(9) 1b ae 4675 
(10) $3 . C1) 24255 
5. (1) 133 (2) 54}. 
(2) 3% (3) 8075 
(3) 1?z (4) 8138 
(4) iL. (5) 11123. 
(5) #6: (6) 121 y5- 
(6) }. (7) 317% 
(7) 236 (8) 9455. 
(8) Ye (9) 6534. 
(9) 3. (10) 5055 
(10) 322 10. (1) 341,37 
6. (1) 225 2) 54TH. 
(2) 188 (3) 67744 
(3) 152. (4) oe 
(Ale. (5) 1048.25 
(5) 148. (6) 15853 
(6) 15. (7) 25568 
(7) te (8) 48635 
(8) 1)P5 (9) Lethe 
(9) les (10) 76012: 
(10) 1.9% 11. (1) 172 
7. (1) 833 (2) 121 
(2) 2548 (3) 147% 
(3) 1433 (4) 135 
Cee (5) 10}. 
(5) 2511 (6) 114. 
(6) 8037. (7) 157. 
(7) 14.3%. (8) 10%. 
(8) 2275. (9) 954. 
(9) 2832 (10) 16,5. 


Boyae spy 


(2) 122. 
(3) ie 


_ 
iv) 
| te NY see o™ 
H OO — 
SE NE) Ww 
NL IR ole et 
pi oojoo ° 
ol? « O) 


(oN 
(or) 
= 
HCO Cl color colo | 
aon Slr Oo ED by} 
e ° ° A ° 


_ 
is 
Fon 
Fo oan an 
Oonr WN KH CO CO 
Neda, Cal Nea Na a el eae ai aa 
“He. IL SR Oo]. col onl bol 
feaeie wl, wien al” cx02 o|? HY” 
SHOC mh eal ies ge 


| 

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A 
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Vea 
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er 

CO em CO NM Cre Or OO 
ee 
i) 
on 


(2) 10;%. 
(3) 12,75. 


(10) 698;. 


18. (1) 3002. 


(2) 4653. 
(3) 6208. 


(4) 76325. 


(5) 8511, 


(6) 18734. 
(7) 25083. 
(8) 46971. 
(9) 84245. 
(10) 7524.9. 


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om 
— 
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mt ID AD wits HH 
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(10) 25,5. 


20. (1) 6198. 


(2) 27's. 


P) 
Ce Se il, bit cole ° 
. ° * wt ° 
r 


(4) 13. 


Pre 

ou 

wa 

bo 
So" 

ol 


ceo co}. 
° 
. 


- RR bd 
© ol? 


(10) 434. 


23. (1) 158. 


(2) 562. 
(3) 641. 
(4) 2568. 
(5) 4283. 
(6) 5452. 
(7) 1153. 
(8) 578, 
(9) 3308. 

(10) 258, 


24. (1) 42. 


(2) 3. 
(3) 49. 


ANSWERS. 


_ 
CO 
Sd 
ms OI 
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| 


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— “~~ [~ N oN 
oS CO =~1 OS Be WwW = © 
Nat TS, NSF ee 
AL O., AL el el LL, th ol. Sie 
of al ol” On, ols WIcO oolax o0|" o0\” oS win | 
as ee ee CC 
[| S| . . . . 


o 
o 


ee, 
RCO or 
| 

. . 


el el oe, 
boco ke OF 

DID ico oo] 

cn Stay = 

° ° 


{| -} ps 
mor loo to 
5 oy Ol oy 


“~~ “=~ 

a= bo 

ae ow 

Hho ot wito et 0 
— a4 

ol aI * 

S| “ 


= 
Or 
ner 
o>) 
) 
_ 


oy 


o~ 
~I 
<—- 
oo colts tl oro ool~r 4] 
* Bin ow ° 
Ptr ae 


DMP OO CiD Cl 


(2) 5152. 


(3) 194.5). 
(4) 66122. 


(5) 1343,5,. 
(6) 198413, 
(7) 16215, 
(8) 8864. 
(9) 6881}, 
(10) 540,%. 
29. (1) 21205}. 
(2) 330752. 
(3) 429971. 
(4) 554628, 
(5) 627703. 
(6) 980991. 
(7) 169,181,%,. 
(8) 319,548}. 
(9) 566,915%. 
(10) 505,896,%,. 
80. (1) $1583}. 
(2) 7083}. 
(3) 106873. 
(4) 36992. 
(5) 51450. 
(6) 58176. 
(7) 11880. 
(8) 57800. 
(9) 352455, 
(10) 25800. 


LON LEN ON ON FN FS EOS 
on oo» Ww & 
ee 
col Re] ol, bol, Col. mL. onlew Colne 
metus oy ti of" 


is 


co 
~ 
“™ 
— ( 
=", 
eH Ol 


y ca. 

le} 

nN 
© oe polar 29)" oe Of 
ps Pea a4 


8 ANSWERS. 


pS 
[@ ¢) 
ae 
Af 
_ 
i} 
oo} i 
oO} 
| ein 
We} 
Ww 
o> 
~] 


(6) 20. (7) 138. 


3 1 473 
9 856° 
(7) 19. (8) 5333 (9) $2633, (10) 136355. 
(8) 423. (9) 1333. (10) $3834%. | 45. (1) £25. 
(9) 27. (10) 144%. 41, (1) 38. (irlhds 
(10) 26,5. 37. (1) 35§. (2) Ys (3) $2. 
33. (1) 1}. (2) 5434. (3) 145 (4) 4°58. 
(2) 23. (3) 7135 (4) 335 (5) 2t. 
(3) 1. (4) 9134. (5) 425 (6) 144s- 
(4) 67%. (5) 1047%5. (6) lis CT) 3343. 
(5) 43. (6) 16234. (7) 33$- (8) 333o3- 
6) 228. (7) 28012, (8) 139. (9) 1424, 
1H : i 1433 
(7) 144. (8) 52953. (9) 175. (10) 383? 
(8) 45. (9) 9405. (10) $63: 
(9) 21, (10) 83822. 42. (1) 6193. Art. 145 
(10) 14%. 38. (1) 441. (262s li 
34. (1) 248. (2) 6431 (3) 61212, 2. $52.30! 
(2) 204. (3) 8183 (4) 47%. 3. $391. 
(3) 113%, (4) 1134 (5) 6023. 4. $445 
(4) 238. (5) 1283, (6) 60235. 5. $0.16498 
(5) 46. (6) 1947 (7) 1497. 6. $4491. 
(6) 24,9, (7) 33182 (8) 50,4. 7. $493.8. 
(7) 1434. (8) 63;%'5 (9) 733. 8. $6. 
(8) 351% (9) 112,88. (10) 42, 9. $19.011. 
(9) 1528 (10) 100182 43. (1) 1113 10. 3127 m. 
(10) 10,%,. 39. (1) $38.5. (2) eS, 
Sees, (2) $8234, (8) 188yy. page hae 
(2) 27435 (3) $9453 (4) 7%. 1 F 
(3) 1335 (4) $84i7 (5) sis 2. 500%; 20%. 
(4) 1308 (5) $603. (6) 382. 3. 40%; 250%. 
(5) 130% (6) $72.4. (7) 38%6- 4. 662%; 150%. 
(6) 2o%%5- (7) $1524. (8) We5- 5. 25%; 400%. 
(7) 2335. (8) $803%%5 (9) 33% 6. 50%; 200%. 
(8) Lies (9) $9420, (10) 25535 7. 1500% ; 62%, 
(9) 1333 (10) $25373. | 44. (1) 38253. 8. 1%; 10000%. 
(10) 55/37. 40. (1) $6y%. (2) 6725. 9. 50%; 200%. 
36. (1) 27%. (2) $817. (3) 86335. 105 Ose 
(2) 1333. (3) $2344 (4) 96544. 100,000 %. 
(3) 224%. (4) $8333 (5) 136333: 
(4) 653%. (5) $6343 (6) 223335. II. 
(5) SPY. (6) B7gy%5- (7) 34430. 1. 400%; 25%. 
(6) 4392. (7) $4377. (8) 881493 2. 335%; 300%. 


S Comin» PP w 


_ 


SCOMWAH AP wWD 


OID oP © WO 


. 250%; 40%. 
. 120%; 831%, 
. 600%, ; 162%. 
. 50%; 200%. 
. 125%; 80%. 
. 60%; 200%. 
. 60%; 1662%. 
. 426% ; 2331 9, 


Il. 


. 500%; 20%. 
. 61%; 1600. 
. 2662% ; 871, 
. 60%; 1662 %. 


500% ; 20%. 
133}% 5 75%. 


. 200%; 50%, 
- 20%; 500%. 

. 100%; 100%. 
. 831%; 300%, 


Vx 


. 2400 % ; 44.4, 
. 700%; 142%. 
. 1200%; 81%, 
. 800%; 331%, 
. 500%; 20%, 
. 1331%; 75%. 
. 300%; 331%, 


Lof 1%; 
20,000 %,. 


. 662%, ; 150%. 
10. 


210/-FA 6 0 
1834 3 048, 0° 


ATt. 153: 


2 66297. 
. 662% Swedish; 


31 0 
51% other 
sources, 


. $1000. 


i) 


, 80% ; 


SCOMWWAATH 


ANSWERS. 


. 8% brass. 


6} %. 

$ 50. 

& or 555%. 

2 or 60%. 

5h 3 


A See Aa 


. 333% ; 6% ; 


121%; $0.99. 


Att. Loi 


. TLA. at 10¢a 


foot costs 
$8963.20 more 
than 560 A. at 
25¢ a sq. rd. 


. 1877 bu; 492¢ 


rem. ; $99.95. 


. $0.095. 
. 50h. ; 1680 m. 


293. 

$ 15.20. 
179! cans. 
$ 11662. 


Peete rem. 
. $2227.98. 


Art. 159. 


. $9364.88. 


23 
. 73: 


378 
. 10255 full 


steps. 
1440 


tions. 


5 
36° 


revolu- 


3 
+ zo M. 


iwi L DY al, 


Tigaleiis 
Mar. 14; 
May 1. 


13 


i a or) 
oe be 


- 
oO 


Pee 
oe ah 


SCOMMHTRPWHNH 


CO 095 52 Oe CRE Soar 


$ 15.52. 
$ 2001.81). 
3512 Ib. 

263 Ib. ; 7999, 
$ 243.52, 


. $1621.92. 
. $698.76. 
. $212.68. 


$ 15.44. 


. $426.65. 


ASC eLOL: 


$ 3.983. 
$2.85. 

$ 13. 
$870.50. 
$ 1.574. 


. $216. 


9073 sq. yd; 
4 squares ; 
36 squares ; 


32670 squares. 


19. 
20. 


_ 


Se oe ee 


) 


1j Tay ie 
89} in. ; 742 in. 
12 in. 


Art. 162. 
$ 58.80. 


. 8800 boxes. 


17s. ; 204d. 


/ ATAU ET Ga! 


nile is 73.3 ft. 
lessthan 1}! xa 
common mile. 


“OO: VOs 520 sable 
5 2 
ee 2 ge eas 


189022 Ib. 


2 


ve 
. $2,819. 


Art. 163. 


. $105. 


1 
+ bu. 


. 59 posts. 


44d. 

Mary, 3; 
Sarah, 4; 
both in 17h. 


. John, 4; his 


brother, +; 
both in 22d, 


. $8000. 
. 80 jars ; 18750 


eggs, 


4 
Ie Aro: 


10. 


20 bu. 


ATLA LOo: 


. $135. 
. Clifford, 


kee 
60? 


Clifford and 


10 


_ 
oO 


OW WWD. HM 


Sos OP ig 3) Cag in ten og Oo 


rol Raila sent tee kL Pil oe 


Leonard, 
Leonard, 
Leonard 


. 
) 


40? 
can 


dh, 


do the whole 


in 40 min. 


lt sis 10/54 


Thursday p.m. 
in New York. 


. 0.04 Friday 


A.M. in Holy- 


head. 
$ 8400. 


. 36 meters. 


$ yd. 
$ 11.25. 


Art. 166. 
16 tables. 


$ 195.12. 
39 bu. 
$ 48. 


. $5964. 


Art. 172. 


100 


or 4910 ft. 
1 meter. 


Art. 177. 


He ele olen | | 
ale ae § al of 
. . . e . 


en 
by 


. The father. 
. 23d. 


yA DOUG as I; 


ANSWERS. 


DoH onion 
; A 


ly Hie BIH 


i 
Poe 


BI) RL, BL, BL, ole BL BY 
al wt al” a" = of af 
Be CR SAT al a 


_j- 
ts 


iH ot 4] 
wi~r ° 
2 


i) 
i> a 
jon = bo 

S| 


co 
w 
J 
or 


Cal iss: 


PArtsee so: 


, 0.588}. 
. 0.8381. 
. 0.5624. 
. 0.2662. 
. 0.4284, 
. 0.5558. 
. 0.2381. 
0ST TR 
. 0.4662. 
. 1.0623, 
. 0.8333. 
. 0.9166. 
. 0.5555. 
. 0.5454. 
. 0.1838. 
. 0.4666. 
. 0.0016. 


25. 
26. 
27. 


oR oO OD HAIHM TARP 


et BE IS peal N NDE | 


0.5888. 
0.2307. 
0.9280. 


Art. 181. 


. 206.568. 
. 335.722. 


21.8725. 


. 22,32589. 


700.108. 


. 680.40235. 


234.696. 
25.1429, 


. 222.585. 
123.27114. 


ATC toe: 


. 226.472. 
. 396.2494, 


152.807. 
3.193. 
23.5237. 


Art. 184. 


1.74. 
1.5056. 
31.13. 
8.774. 
3.749, 
0.3976. 
L767; 
1.5306. 
27.926. 


. 5.87. 
. 5.893. 
. 2.5416. 
. 23.8352, 
eeO-00 Ls 


Art. 185. 


182.62. 


. 2.886. 


= 
f=) 


—_ 
i=) 


CA and pid Bode Os ORAS oe ots, an 


ENE Pd Mie 


BORER ON CoC bn 


1.775. 
0.162, 
0.0162. 
7958.2, 
0.235. - 
6.0. 
1.62. 


. 5.562. 


Art. 186. 
2.975. 


Aiea 7c 


99.75. 
0.501. 


ree0es 


Art. 190. 


. 79.88904. 
. $8.064. 


6.963744. 
4.23, 
0.46875. 
0.675. 
46.656. 

$ 480. 
51.20. 


A Ag Sate 


Art. 191. 


$13.49. 
$ 16.92. 
$ 1050. 
1372.5 T. 
$ 8.55. 
520.2. 
750 Ib. 


. 0.5625. 


$ 80. 


. $6000; $120. 


_ 
Oo 


td ir haha gel pg Aine At ee 


iP go gO 


T0 
mn Lt. 


Art. 193. 
3600. 


. 0.289. 


78.4. 
0.39. 
70.5. 
113.5. 

$ 8.9125. 


. 640.0287. © 
. 55.1286, 
. 8560. 


Art. 196. 
14s. 


| bole 
° 


16 
00° 

is 0.04d 
more than 
£0.004, 

50 %. 

2 doz. 


. 48 sheets. 


5%. 


. 61%, 
. 15%, 


1.12. 


Art. 199. 


. $89.25. 
. $89.25. 


10400, 
(st ward ; 
12800, 

2d ward ; 
8400, 

3d ward ; 
8400, 

4th ward. 


. 9159.2 ft. 


en ee ee 
Seomont our wonr oO 


so 


eee 


$05 O05 Es Ge ON ee 002 Sos 


ANSWERS. 


, 258 balls ; 


1082 oz. 
175. 
0.049. 
0.025. 


. 0.33338. 
fs LS tet 


0.183; 4%. 


Art. 200. 


. 2.005. 


0.0831. 
0.081. 
0.10. 
0.68. 
0.075. 
0.0125 ; 4. 
0.125 ; 4. 
0.0025 ; 


a 
400° 


. 0.2124. 
Z4.0.0t 

. 0.0625. 

. 0.015625. 
. 4g). 
Pieced 

. 2.25 ft. 
, 18it. 

. 2 sq. it. 
a1 Sisq. it. 


‘ATE 20. 


. 208,876.8. 
. $0.75. 
. 5500 Ib. 


4750 lb. 

10334 Ib. 
15040 Ib. 
35624 lb. 


. $7.20. 
, 21%, 


of the 


men receive 


10. 


© @ = 


0.15 more than 
$1aday; the 
rest 0.388 more. 


. 0.081; 0.125; 


0.002723. 


. 0.638341 yr. 
. Sept. 29. 


$ 1. 


. 13} loaves; 3. 


Art. 202. 
eee Lie 
128’ 128’ 
52 | 
128 


. 0.8089, 
. 0.5218 ; 0.071; 


0.052, 


. 0.3771, 
. 0.0075 ; 


0.0028 ; 0.9990. 


0.183. 


yO! 


Art. 203. 


1 


B° 
Oe ie 
. 8453.775 ft., or 


3756,225 ft. 


4. 1m. 

5. $180.60. 
6. 
7 
8 
9 


124 sheep. 


. 90 sq. in. 


- yr. 
. 25m. 


64 ed. 


oo 0 


11 


Art. 204. 
. $1840. 
yt ee Wt WF 
78.0195. 


on 


> 


CO BR aH Dip Oo} 
Om of Hl 
mM 
OQ + 
= 
— 


Or Hy 


=~] 


(520.977. 


Lh P 
Otek hoe 


ee 


1. 252 d. 
2. Lata; 607%. 


4. 


OHrHys 


. $38.88. 

1 $20-bill ; 

1 $10-bill ; 

1 $5-bill ; 

1 $2-bill; 

1 $1-bill; 

1 half-dollar ; 
1 quarter-dollar; 
1 dime ; 

3 cents. _ 

11 pieces. 
£15; 


795 er. 

7 oz. 4 dr. 
. $0.212. 
$412. 

4 OZ. 
15 gr. 


1 pwt. 


Art. 206. 

. 3216} m. 
463°. 

. 691m.; 144m,; 
1014 ft. 


10. 


17,5 m. 

720 ft.; 690 m. 
37 double 
eagles. 

20 ecu. ils 
bi CU sin, 5 
113} gal. 
537.604 cu in.; 
41.85 + bags. 
48 pt.; 80 gi; 
2 pk. 4 qt. 
20; 17 cwt. 50 
IG, eae pk: 
SSG peel Dt 
4 gi, 


Art. 208. 


147. 
87. 
64, 
1000. 
67.67. 


Art. 209. 
$4. 
1.88. 
LEYS 
Oe hoe 
9.33. 
ooreG. 
45. 
11.68. 


Art. 210. 

$ 26.55. 
134.81. 
127.80. 
12.51. 
90.71. 
81.58. 


© © 


10. 


oI wo 


Qa Reve 


ANSWERS. 


«epi OGs 


195.70. 
21.08. 
8.09. 
188.85. 
50.25. 
16.67. 
31.06. 
51.79. 
526.46. 


Att; 214; 


1 
z m. 


62 it, 


. 2455 m, 


Pat Oat b 
6215227 m. 
Ist rider, 2 m. 
in vi mins 

“ 15 
2d rider, $3m. 
We Rnbah Aa 


. 890154 ft. 

. Qrd. 918 ft. 
. 6145 m 

10. 


37632 revolu- 
tions. 


Art. 217. 


. 140 sq. rd. 
. 640 A.; 64.A.; 


560 A. 
36 sq. m. ; 
24 m. 


Art. 218. 
1588 sq. ft. 
45 479, sq. rd. 
781854 sq. ft. 
864000 A. 


. 43560 sq. ft. 


PLUS SSC 


Oana P w Ww 


. $3411.50. 


3821) A, 


. 135935 sq. rd. 
. 176 sq. ft. 35 


eqyins 
$ 5250. 


Art. 223. 


AV PANE (e fae VER 
. 324 sq. in. 


288 sq. ft. 


260 sq. ft. ; 
272} sq. ft. ; 
504 sq. yd. 


fee A. 
if 
. 403 sq. yd. 
. 84 sq. yd. 
10. 


1020 sq. in. 


2916 squares. 


Art. 224. 


$17.50. 
389 qt. 
4 rd. 
50¢. 


. 576 tiles, 
. $12.50. 
. 3840 pieces. 


Art 2225, 


. Carpets 1 yd. 


wide could be 
used on floors 
L2Sth el ete 
27 it...and: 18 
ft. wide; car- 
pets $yd. erie 
on floors 224 
ft. and 154 ft. 
wide. 


ee a 


On Powe 


Neither width 
could be used 
on floors 20 
ft. wide. 

be strips; oo 
yd.; 374 yd. 
7 strips; 49 
vd.; $61.25. 
$ 25. 

$ 70. 


. $26.67. 


$ 28.35. 


. $258.90. 

. $24.83; 

- $90.92. 

. 21 tiles: 

. 20924 tiles. 


3245 sq. ft. 


. 1048 tiles. 


21,7; sq. ft. 
770 tiles. 


-- 14-rolls: 


' Art. 226. 


. 2850 slates. 
. 261386 blocks. 


6212 plates. 
13 at ft. more. 
$ 147. 


: 48° sq. Lhe 


200 sq. ft. ; 
64 sq. ft. ; 
LO SO sat tas 
140 sq. ft. 
712 sq. yd. 


Art. 228. 


. $4098.60. 


300. 
937.50. 


4. $1350. 
§. 2559.38. ! 
S © 1728. 
%. 2245.32. 
8. 1975.59. 
9. 2602.33. 
10. 1447.88. 
ii; 937.20. 
12. 1929.60. 
13. 272.68. 
14. 604.08. 
15. 407.15. 
16: - 94.08. 
Ling ool 
18. 67.50. 
19. ~ 124, 
20. 320. 
Art. 229. 
7. 93% sq. ft. 
8. 10560 sq. ft. 
9. 11} sq. it. 


. 3& sq. it. 


Art. 231. 


- 260 sq. ft. ; 
110} sq. in. 


. 1614 sq. ft. 


10. 10/5 sq. ti, 
Art. 232. 
5. 180133 sq. ft. 
6. $25. 
Art. 233. 
10. 360. 
11. 750. 
12. 11 sq. ft. 
18. 231 sq. ft. 


. 73} sq. ft. 
. 69753 sq. ft. 


ANSWERS. 


Art. 234. 


. 896 Sq. in. 
. 7500 sq. ft. 


Art. 235. 


. 62.832 ft. 

. 66.25365 ft. 
. 665.488 ft. 
. 1.485442 ft. 
. 61.051 in. 

. 1.195664 ft. 


Ia, Peel oe 


. 60. 


113.0976 sq. ft. 


. 78.54 sq. ft. 
. 795.775 sq. ft. 


198.94575 sq. 
ft. 


Art. 237. 
19.635 sq. ft. 


1 


4° 
113.0976 sq. in. 
$ 8.48. 


. 51416 sq. ft. 


Art. 238. 


. 1256.64, 


509.296. 
1809.5616. 


. 1017.8784. 
ts) Rote 
. 1963.5. 


Art. 241. 


. 16+ in. 
RABE 
. 4.00009 + rd. 


4. 


oO 


14. 


. Curved 


OHARA P ww 


. 600 cu. ft. 
10. 


Diameter of | 
hhd. = 3.978 
+ ft. ; 
Diameter of 
door = 3.833 
+ ft. 

1.8169 m. 
edge 
eon AC) 
Straight edge 
= §7.2958°. 


. 18.8496 in. 
. 496.24 ft. 
. 9685.84 sq. ft. 


Art. 243. 


216 cu. in. 


J eoL2: Gus 1D; 


ip Om it 


Sly 2S cu.ft, 


1000 cu. yd. 
8000 cu. in. 
125; 348; 
12 © 12S 
216 ; 1000; 
133 ; 8000. 
Aas PADS S 
TG oo. 


Art. 244. 


. 180 cu. in. 


400 cu. in. 
74 cu. ft. 
1280 cu. ft. 

4 cu. ft. 
4608 cu. ft. 
115831 cu. ft. 


1772 cu. ft. 


10. 
Re 
12. 
13. 
14. 
15. 


C2) Sie Bee 


10. 
i 


13 


Art. 245. 


$ 18. 
100. 
112.50. 
19.69. 
29.30. 


5.50. 


Art. 246: 


10 sq. ft. 
120 sq. ft. 
12 boards. 
2Osbdy dies 
15 bd. ft.; 
123 bd. ft.; 
20° bd. Tt.: 
a >< 1LOsbas Te 
ove bd. ft. 
256 bd. ft. 
131 bd. ft. ; 
162 bd. ft. ; 
20 bd. ft: ; 
21% bd. tts 
231 bd. ft. ; 
15 bd. ft. 
110 bd. ft. 
537% bd. ft. 
216 bd. ft. 


Art. 247. 


. 486 cu. in. 
zo GOOLEn. in, 
. 1536 cu. in. 


8 in. 
10 ft. 


eer hall: 


Art. 248. 
114 sq. ft. 


. 88 sq. ft. 


14 


OOIP A 


10. 


© © FP wo WO 


. 0.2146 


. 248 sq. in. 
. 468 sq. ft. 


1240 sq. ft. 


. 8485 82. 
. 552; 973. 
. 576; 224. 


Art. 249. 


shav- 
ing; 
0.7854 
der. 


cylin- 


oh G elie ins 


7854 . 7854. 
10000 %9 10000° 
31.416 cu. ft. 

399.2928 cu.in. 


. 602:656 cuit, 
7 O28.562-ClLeLG. 


ian? 


Art. 250. 
12.5664 in, 


. 2.54648 in. 


25.1828 sq. in. 


. 314.16 sq. in. 
. 353.43 sq. in. 
_ 224.3456sq.in. 


Att ecoe: 


Rab aii eC se alas 


perimeter 30 
ee 
S) oli oo dll yes 
perimeter 26 
ing 
porte. Pe tehiee 
perimeter 24 


in. 


. 27.225 ft. 


ANSWERS. 


3. 12 in. average | 8. 30 in. 
width ; 9: 11-ft. 
200 running | 10. 2.8 + ft. 
feet. 11. 4 ft. 
4. 13} ft. 12. 660 ft. 
§. 1.2784. 
7. 3 in, Art. 255. 
8. 128 oranges. 1. 3 ft. 
10. 800 A. 2. 1840781 gal. 
3. $48595 gain, 
Art. 253. | 4. $ 133650. 
ey ies 
1. 24x6=144;] 6 op 
Me Sera Wt Re 7. $2.50, 
ae suit 8. 960; 6. 
ae 9. $135.41. 
6.- 78 sq. ft 10. $9. 
pate 11. $2242.50. 
8. 120 ft 
9. 9 ft. Art. 257. 
10. 5443 ft. 1. $50.63. 
eA, 2. 109394 Ib. 
12. 18 ft. 3. $26.98. 
et ee 4. $1505.44. 
1420 rd. 5. 37.6992 bbl. 
ee INE 6. 603.1872sq.in. 
16. 62500 sq. ft. 7. $276.67. 
17. 31.416 ft. 8. $5760. 
18. 7854 sq. ft. 9. $476.80. 
oe ne ft. 10. Perimeter of 
20. 6 in. 6-inch square 
Art. 254. Ofc oneeleree 
1. $10. tangle =26 in.; 
2. 4d. of 3 x 12 rec- 
3. 6 in. tangle = 30in. 
4, 12 ft. 
5. 12 in. Art. 258. 
6. 4 it. 1. 0.2146. 
7. 1089 ft. ; 2. 909. 
5°, tt, 3. $6106.88. 


Oop 


170, sq. fi, 
1,5 % loss. 
$0.13. 


. Once in 1000 


times. 


. $17.82. 
. 630. 
. 83% min. 


Art. 262. 


. $5376. 
. 4801 bu. corn; 


3542 bu. oats ; 
4807 bu.wheat. 


. 9354 T, 
. 68.8 m. 
. 9552 A. 
. $2.44, 
. $209.25. 
. $61.19. 
. $400. 
. $56000. 
. $560. 
. $434. 
20. 46946. 


Art. 265. 


ool Au 
. 6300 volumes 


in library ; 
1764 works of 
fiction. 

(Aes 

96%. 

30000 T. 


. $17634.37, or 


453%. 


824), 


. $106.25. 


444 9/, 


ee ee 
a Pp we 


ee 
See Pee Oak Ne We COLD” bt 


Art. 266. 


Aliare equally 
profitable. 


. 2ofsomething 


is 100 x 2% of 
it. 
25 %. 


. 581%, 
. 45%, 


95 %, 
162 %, 


. 4; 800; 2% 


7 3 


. 6; 1600; 1%. 
. 112; 


83} i 
163 yee 
13; 140; 3 


3 


2910). 21. 
: 331%; 31; 


25 Y, 


- 625; 2000; 


1000. 


. $1241.67. 
. $ 12000. 
. 864%. 


Art. 267. 
12, 
960. 
16%. 
40 A. 
92%, 

$ 1728, 
564%. 
225 cd. 
7800 T. 
4%, 


. 18%. 
. 138%, 


- $9600. 


5 
re 


10. 


SIAR ewe 


ean eee ots an DO ars 


oP oO De 


ANSWERS. 


Art. 268. 


. $6500. 


$ 24640. 


- $60; $72. 


331 94, 
162 %, 
25 9, 

33! 9, 


LO ] 
. 31% more is 


gained by 
buying for $4 
and selling for 
$ 4.80. 


. $9. 
. 81%, 


Art. 269. 


$ 4.80. 
$72. 

$ 6000. 
(pk 

500 A. 
$ 50000. 


. $2000. 


60. 
35%, 


Art. 273. 
80 %,. 
$ 60000. 


. $50000. 


$7.81. 


- $5000. 


1B ie 


co 


oO ROOST CAC, teas ro Oates 


Art. 274. 
$ 96. 
$ 48. 
$ 32. 
$ 30. 
$ 30. 
$ 35. 
25 Yf,. 


. $20 or 25%. 
. 20%, import- 


er’s gain ; 


162%, retail- 
er’s gain. 
142% 

$870 

Art. 275. 


. 183% gain on 


meat; 142% 
gain on pota- 
toes. 


. $10. 
a 3 
. 1g saved by 


buying to-day. 


. $48000; 


$ 50400. 


. $140. 
. 15E 9%, 
. bY, 
east he 
. 60%, 


AS TE0 
147% % 


. 84, 
. $5200. 


Art. 277. 


. $44.80. 


137.60. 
28.05. 


15 
6. $5.88. 
if 298.54. 
8. 3785. 
9. 104.99. 
10. 125.16. 
Art. 278. 
(oe Re Ree 
8. 9.92. 
9. 1.56. 
10S is 
lives tea: 
192 seis 
135 720/80; 
14570211653 
15. 38.65. 
16. 0.05. 
Art. 279. 
9. $4.49. 
Die 10.672 
12. 15.53. 
133 4-13.27. 
14. 14.21. 
15. 6.438. 
16. 1.90. 
17. = 4.09. 
18. 9.06. 
190 ec: 
20. 8.68. 
21. 196.67. 
pV Wed laste 
9300 16716: 
Art. 281. 
3. $546.86. 
4. 7920.11. 
5. 1909.96. 
6. 95.06. 


16 
7. $1135.24. 
8. 314.28. 
Ors .02.02; 
10,4 -11S8tt26; 
11. 4838.69. 
Art. 282. 
(a prye'4 ies 
8. 2 yr.2 mo. 6 d. 
9. 2yr.10mo.13d. 
Art. 283. 
1. $14.49. 
2g. fe ee: 
feet ee sO! 
4. 7.99. 
i a af PavAs 
6. 937.50. 
Nee calouw es 
8. 461. 
9. 21.85. 
10. 166.45. 
1. 1.89. 
127 6 1 
13. 49.80. 
LSPS thee 
Art. 285. 
1. $26.79. 
2. 759.54. 
8. /640.11. 
ANP51506; 
5. 900. 
6. 540.02. 
Te 9234, 
8. 60.67. 
OF 202205 
10. 3.15. 
phar ve 
PR By eye 


ANSWERS. 


18. $147.52. 
14. 4.41. 
15. 19.44. 
1Gseee 0: 
17. 58.90. 
18. 227.50. 
192% 746.67; 
20. 105.22. 
21. 385.33. 
O22. 20: 
23, 4.25: 
24. 48.95. 
Obs viisl2.50: 
Art. 286. 

4. $144. 

6. 4.93. 

71o0..0,00. 

Soe ies G: 

9. 11.05. 
1016.7 72 
11. 7:29.59: 
12. 160.98. 
1353 coc t.Gia 

Art. 289. 

5. $1900. 

6. 10%, 

7%. $500. 

8. 4%. 

9 $1.50. 

10. $500. 
Art. 290. 
12s los 

Bre OU Uae 

3. $0.40. 

4. $0.593. 

5. 50% of $14 ; 


20 9, of 50 ¢+ ; 
1% of 40¢. 


— 


SHMARBAP wD 


11: 


$ 11.70. 


. $24.26. 


$520.10. 
$ 5.24. 


. $240.88. 
. $2390.08. 


Art. 291. 
$ 6500. 
19, 

40. 


. Cash discount 


is 3%. 
$ 1266.67. 
$ 2375. 


- $1400; $1425. 


20 %, 


. $ 10472. 
. $8920. 


73, 


. 15%, 
. 20%. 

. $313.63, 

. $10423.88, 
. $598.50. 


Art. 293. 


. $10000. 


$ 10000, 
$ 2000. 
£%: 
119, 

$ 4000. 
$ 62.50. 
144. 
$31.60. 
$ 5000. 


30/7. 17 
: o%5 40° 


$ 5625. 
$ 375. 


Do P oo 


© oc =-t 


12. 


Ree Ly 
2d risk is 1% 
cheaper. 


. $230.09 ; 


9 8 a % . 


. $2040.82. 
. $7000. 
. $120; $3388}, 


Art. 294. 
80 ¥, 


. $110; 4%, 


$ 8.10. 


. $6.91. 
. Of gross re- 


ceipts. 


Art. 296. 


. $120. 
- 1023%; $20; 


$2000. 


ee yee 
. $1600; $1620. 


400 


£07° 
Art. 297. 
103 ; $27. 


. $1200 amt. in- 


vested ; $1242 
remittance. 


. $34.95. 
. $2.50; 15%. 
. $293.06. 


$ 8.74 ; 
$ 428.26. 


. $2750; $2675. 
. $25; 
. $1000; $20. 
10. 
1}: 


10%, 


14%; $1082. 
$ 1552.80 ; 

$ 67.20. 

$ 450.70 ; 
3%. 


pl 


oO 


ae 


Fete ee oe 


| aioe 
y © 


Art. 298. 
$421.05. 
526.67. 
200.00 
9,65 
4.57 
$ 214.00 
20000. 
589.20 ; 
112% +. 
3.52 ; 
2.37% +. 


777.65 ; 3%. 


207.90 ; 
198. 
1250 ; 
45.75. 
787.49 ; 
15.39. 


Art. 299. 


$314.95. 
12000. 
4123.38. 


7600 ; $400. 


16.87}. 
9250000. 
4006. 
30000. 
6207.63. 
58.87. 


Art. 301. 
$ 261.52. 
987.60. 


76.58. 
1356.55. 


1847.71. 


10. 
1b 


10. 


je 


Se tie as Se 


ooh 


ANSWERS. 


Art. 302. 

$ 355.80. 
600.60. 
370. 


Art. 304. 
$780. 
3180. 

576.31. 


Art. 305. 

$ 156.10. 
865.49. 
2280.01. 
750.97. 
197.74. 
806.99. 


Art. 306. 
$45. 
$45. 


. $128.65 


Specific duty 
$40 more. 


. $60.75 ; 34%, 
, 1221%; $1650. 


$ 132.60. 
20 %, 
15 %. 


. $0.32 gain. 


Art. 307. 

$ 300,000. 

$50,000 ; 
10,000. 


$150n $1000; | 
14 ¢ on $1; | 


$ 8000. 


13%; $0.012; | 


14%. 
112; $1.48. 


iO 


0407 SRE Py 


. July 7. 


10’ >? 


$ 120.88. 


Art. 308. 


. $78.81. 


206.46. 
449.95. 
1030.30. 
38.14. 


Art. 309. 


$366.94 +. 


424.48. 
270.61. 
812.06. 
428.85. 


Art. 310. 


. Mar. 3. 


Sept. 8. 
May 31. 
Mar. 3. 


. Nov. 80. 
 Feb.-22: 


Feb. 28. 
Apr. 18. 
Nov. 


Mar. 3. 
Apr. 29. 
Art. (SOLE: 
$ 590. 
1187.60. 


7 per SLOVO; | 


10. 
un. 
“12. 
13. 


14. 


17 


Art, 612, 
$ 790.70 
(grace) ; 
$791. 
$7135.70 
(grace) ; 
$ 714. 


. $8.97 (grace); 


$8.75. 


. $594.75 


(grace) ; 
$ 595. 
$ 523.56 
grace) ; 
$ 523.69. 
$ 906.54 
(grace) ; 
$ 906.68. 
$ 310.82 
(grace) ; 
$311.04. 
$ 784.50 
(grace) ; 
$ 785. 
$ 953.28 
(grace) ; 
$ 953.60. 
$ 715.58 
(grace) ; 
$715.74. 
$ 836.25 
(grace) ; 
$ 836.59. 
$ 870.90 
(grace) ; 
$ 871.05. 
$ 91.32 
(grace) ; 
$91.54. 
$ 5907.75 
(grace) ; 
$5910. 


18 


15. 


16. 


ma IE Pag ine! A 8 ag ae 


$ 4241.50 
(grace) ; 
$ 4243.29. 
$ 814.93 
(grace) ; 
$ 815.43. 


Art. 313. 


. July 7/10; 


27 d. or 80 d. 


. Feb. 15/18; 


63 d. or 66 d. 


= Octibs3- 


37 d. or 40 d. 


. June 17/20; 


47 d. or 50 d. 


2 Aug.4 77. 


57 d. or 60 d. 


. Feb. 24/27 ; 


26 d. or 29 d. 


. Jan. 19/22; 


70 d. or 73 d.; 
2 mo. 9 d. or 
Foe Le eee 


. May 11/14; 


66 d. or 69 d.; 
2 mo. 5 d. or 
2 mo. 8 d. 


Art. 314, 


. 63 d. or 66 d. 


24 d. 
$ 438. 

$ 994.86. 
$ 795.33. 


. $2980.50. 


$ 447.65 


(grace) ; 
$ 447.80. 


$ 705.06. 


10. 


14. 


15. 


16. 


it: 


ANSWERS. 


. $794.22 


(grace) ; 
$ 794.55. 


$ 1195.50 
(grace) ; 

$ 1196.45. 
$1195.97 


(grace) ; 
$ 1196.69. 


. $247.60 


(grace) ; 
$ 247.69. 
$ 1716.07. 


. $188.74 


grace) ; 


- $188.80. 


$ 273.24 
(grace) ; 
$ 275.48. 
$ 275.18 
(grace) ; 
$ 273.87. 
$ 345.44 
(grace) ; 
$ 343.70. 
$ 752.65 
grace) ; 
$ 752.85. 
$ 752.78 
erace) ; 
$ 752.97. 
$ 4988.89 
(grace) ; 
$ 4989.93. 


Art. 315. 


. $1223.64. 
. $502.24 


(grace) ; 
$ 502.15. 


| 9. 


dined rant iD AAR et La 


. $44,166,666.67 | 
p25 | 


$ 726.92 
(grace) ; 
$726.81 (time 
inexact days). 


Art. 316. 


1 


. $60; $8. 


Art. 317. 


. $5; 5%, 
85. 


$ 1125. 

$ 20. 

60 shares. 
$ 101.50 ; 
$ 1.50. 

$ 370. 

$ 100. 
Go, 

6%. 


. $3450; $138; 


4%. 


. $5; $200. 


Art. 318. 


$ 500. 
$700. 
5, 

$ 4000. 
150. 
1102, 


- $10100. 
10. 
11; 


$ 4906.25. 

$15 nearly on 
a 100-dollar 
bond. 


Art. 319. | 


90 9, 


130 %. 


. $2838.38. 
. $9712.50. 
. $4015. 

. $4280. 

. $3940. 

. 8.84%, 

. 549%. 


Art. 320. 


. $476.19. 


694.44. 
1045.48. 
697.12. 
83.50. 
753.33. 


Art. 321. 
2 yr. 


Bday 

. $200. 

. 4yr. 6 mo. 
. $840. 

. Be, 

. $960. 

Rare A'§ veces tele) 
. 43%. 

. $179.28. 


Art. 322. 
63 ¢. 


. 80%. 
. $20.76; 30%. 


Art. 325. 


oy AA AS 
. $6.25; 


$ 1494.25. 


. $1485.50. 
OO: 


after 
date. 


11. 


12. 


——s 
Pa Sag eed ee a Ok os eee Berar 


_ 


12. 


13. 


14. 
15. 
16. 
17. 
18. 


. $3920. 


. M. 3640. 
. $387.60. 


Oop o wo 


. $73.46. 


. $1266.25. 


- $87.70; 


$ 493.50. 19. 
. $490; $489.75 | 20 


grace). 
$3958 (grace); 
$3940. 
$ 10.25. 


£ 2010. 
£ 860 10s. 


Art. 327. 


. 2872.50. 


$ 21.56. 
217.01}. 
2085.71. 


$ 6185. 
$ 12.09. 


$5.19. 
$ 40.41. 


_ 
oO 


$ 89.24 
(grace). 
$ 49.61 ; 
$51.14 
(grace). 


— 
wo 


$ 195.95 13. 
14. 


(grace) ; 
$ 194.74. 
$ 289.59. 
$ 58.80. 

$ 256.10. 
$ 845.56. 


$ 941.11. 15. 


OOH oP 


OHMIRH HP ww 


ANSWERS. 


$ 1886.70. 
$ 1213.21, 


Art. 328. 
$ 250. 


. $4761.90. 
Art. 326. 3. 


Eig AB aT) 
200 /o OF 


0.605 %. 
$ 156.25. 


. $33.39 loss. 
. 6535.95— bu. 
. $4189.47. 

. $1590. 

. Gain $ 14.25. 
10. 


$ 1960.78. 


Art. 329. 


. $402.38. 
- $751.82, 


$ 9.93. 

$ 563.20. 
$3.35, 

$ 820. 
24 yr. 
8%. 


. $905.66. 
- $690.08. 
. $497.78; 


$ 497.57 
grace). 
$ 212.24. 


$48; $47.36; 


$ 48.72 ; 
$ 48.00 
(grace); 
$ 48.30 ; 
$44.44. 
$ 20. 


5. 


. 59 


re) 
, ae 


ob et Ph 


Art. 330. 


ok es 
. 300. 

Se HE 
pee 


1 


#10 Vd. 
. $20. 


75. 
240. 
1536, 
65. 
43.5. 


Art. 331. 


. $15. 

. $378. 

. $126. 

. O¢h. 

. 223d, 
Beebe lt, 
. $133.35. 
ee TAS 

. 62 02. 

teat UeoU, 


Art. 332. 


ye | 


$ 1200 ; 
B. $300. 


. S. $560; 


B. $140. 
$2000 ; 
$3000 ; 
$ 4000. 


$300 ; $450. 


. $10,000. 


| 10. 


rhe 


12. 


13. 


wwnwnnnnndw wnnrerere 
VKH SCHoHKBNTATHEONMOORA 


il 
or © 


13. 
14. 
15. 


Se eS | Sli Ot. Ns OB. BO be 


$ 0.844, ; 
$ 507.70. 
$7411.76. 
$ 6000 : 

$ 4000 ; 

$ 3000. 


$ 20,000. 


Art. 335. 
EE 
Fee: 
, sok 
. 42, 
. 46. 
. 54. 
68: 
. 68. 


75. 


. 84, 
= U4; 
. 73. 
4 On. 
» 92: 
. 58, 
. 99. 


Art. 336. 


532. 
547. 
636. 
746. 
869. 
1462. 
0.75. 
0.96. 
6.5, 


. 0.88484. 
. 0.9485+. 
. 4.41214. 


28.7210+4+. 
4.12734. 
0.8. 


19 


20 


16. 
17. 
18. 
19. 
20. 
21. 
22. 
23. 
24. 
25. 
26. 
27. 
28. 
29. 
30. 


- 
rOoOMBADTP WD 


Fo oe 
SODARAH WD 


12. 
13. 
14. 


0.2529-+. 
43.9590-+4. 
27.9991+-. 
15.03. 
1430. 

8.74. 
0.89444. 


0.559+-. 
23. 
2.380+. 
9.099-+. 
(),5422+. 
Lo o2oor 
1.4790-+. 


Art. 337. 
307. 
0.85384. 
3.7249+. 
1004. 
3136. 
7921. 
45.09. 


. 0,.9682+. 


11. 


. 999. 


Os. 
64. 
4a 


i 44 jee. 
, 0.193642: 


11.2745+4. 


. 28.0178+. 
. 1856. 


530.25244+. 


. 0.860815+4. 


Art. 338. 
44, 
21.9094. 
29,9334. 


ANSWERS. 
15. 61.8464. Art. 342. 
i sels bene a 1. 36 sq. ft. 
Ti 180.00. 2. 2937 cu. ft., or 
a 29.629+ cu. ft. 
19. 186.051+. 4. 471.4 cu. in. 
cata SUL EEE 5. 500 cu. in. 
917 47745 
Art. 343. 
Art. 339. 3. 384 sq. in. 
1. 2 ft. 4. 412.300 cu. in. 
2. 40rd.; 160rd.| 5. 4 sq. ft. 
3. 24.1664 ft. 10. 105 sq. in. 
4. 21.9814 ft. lis 375 sq-sin: 
5. 18,32 esq: tt. 12. 255.62 sq. in. 
6. 4ft.2.9+4 in. | 18. 301.593+ sq. 
4. 20.784. ft. in. 
8. $141.42: 14. 463.624 sq.in. 
9. 15.8664 ft. 15. Contents. 
10. 108.1662+ sq. | 16. 0.2146. 
ye Art. 344. 
Aaa: %. 452.3904. 
8. 12,566,400. 
1 G69282 24-39. 9. “Bd Gee 
‘ : Qs Jit. 
Ce 10. 200,000,000. 
3..5:4. 
4. 21.089 rd. Art. 345. 
5. 208.71+ ft. 7. 113 cusin: 
6. 97.616+ ft. 10. 4.1888 cu. ft. 
7. 63.639 ft. 11. 4,188,800,000. 
8. 176.568+ rd. | 49 0.4764: 
OF 1427 arden 0.5236. 
235.5+ ft. 
10. 101.98-+ ft. A THEaaG: 
9. $2700. 
Art. 341. 10. 4=9: 
by, 1178-10; TVs 1:50. 
6. 670.208. 12. 194. 
4. 942.48. 13: oh 
8. 11.3184 ; 14:03 7876: 
189.554. 15. $21.09. 


OHARA 


Dorp owe 


Art. 347. 
64; 8. 

20 lb: 
1171.874 bu. 
96 h. 


Art. 348. 
$ 9.25, 


. 681.7875+4 sq. 


it: 
167d. 


. 423 sq. ft. 
so LOSdr ite 


$ 32.26. 
124.686. 
875. 


. 1.273824 ft. 
AST 8 


768 sq. ft. 


. 4.1888 cu, Tt. 
. 14.499 sq. ft. 


Art. 349. 
1080. 


. §.2832 cu. in. 


12.65+ in. 
530.145 sq. in. 


. 24902.18142+. 


180 rd. ; 
169.68+ rd. : 
150.40— rd. 


. 94.814. 
. 26.5294. 
a 4:37. 


1607.8125 Ib. 
14.9354+ ft. 


. 28.2744. 


18,000 Ib. 


14. 
15. 
. 2108. 


Ses 


= 


ae ee ee) 


706.86 sq. ft. 
70.71+ ft. 


Art. S65, 


11352, 

$ 63.28 ; 
$ 2763.88. 
$ 59.42. 


1349 cu. in. 


. 274 yd. 


$2.24. 


. 499842, 


24 


605° 


Art. 366. 


$ 1504.29. 
$ 2745.76. 
Neither. 
6750. 

$ 320. 
16.96 % ; 
$16.11. 


Loss $13,351. 


Cash. 

113. 

38377. 
Art. 367. 
120° E. of 


starting point. 


913 
b5}3. 
88. 
821 
1372 


& 


ANSWERS. 
5. 2 min. 3932) 4. 20°. 
sec. 5. $527.15 gain. | 
G.lnteal 6. $3.24. | 
7. 10722 + q. 209. 
8. 300 Ib. 8. $12000. 
9. 16. 9. $6.75. 
10. 8.24. 10. The latter by 
Art. 368. li. 10, 
1. $552.49, 12. $1045.45+. 
2. $2687.44, 13. 18.8%. 
3. $20. 
4. $27.61, Art. 371. 
5. $4378.79. gS a ae 
6. 1413.72 sq. ft. | 92 Int. for 4} yr. 
7. 460.195 + lb. to: 
8. 1583.34 +. $F L380 2m. 
9. $92.36. 4. 389.70+' sq. ft. 
10. $337.27. 5. $100. 
6. 2.652. 
oe SALE 7. 6.6338-+ Ib. 
1. $4372.60. 8. $450. 
2. 173%. 9. $ 107.623. 
3. 10.63-+ in. 10. $789.78. 
4. 35%. 
5. $141.20. Art. 372. 
6. $421.85. 1. 92¢. 
Ue he hs Q. 2714, 
8. $108,800,000. | 3 & 1980. 
9. 63.8 : 36.2. 4. 4129, 
3 /0 
10. $0.01375+. 5. $2924.10, 
11. 752.52— ft.; | § 9 409, 
twice as much 7. $ 145.90 +. 
appa 4’ 8. 17.72 + ft. 
12. fae eae | 9. 1232 cu. in. 
tsar OAS: 10. A $ 10800; 
| B $7200. | 
Art. 370. 11. $851.27 + or | 
1. $217.50. | $851.59 +. 
2. $11.55+. 112. $83}. 
3. $5.91. 13. 9.36 + %. 


14. 
15. 


25 %. 
Lan: 


Art. 373. 
$ 26.25. 


» $256000. 


36 shares ; 
$50 rem, 

20 

62 %. 

$ 3668.75. 

5 0 

5% 

$ 178.76. 

$ 20937.50. 
Increase $ 15. 


. The former by 


5 
$ 10250. 


. $6300. 


75° East, 
10.89 + in. 


. 93.55 — sq. in. 


84.82 + sq. in. 


Art. 375. 
1. $9.5. 
2. 874%. 
3. $2400. 
4. 160 bbl. 
§. $ 58500. 
6. $1980.43. 
%. A$12;B$?o: 
C $24. 
8. 623 A. 
9. 444 ft. 
10. $9.21. 
Lt. 31-66 36: 
12. $1440.46. 
Art. 376. 
1. 50%. 
SB. .12.i6. 


99 ANSWERS. 

3. 614. 9. 40¢. 5. $538.77 ; 

4. 7360 bu. 10. $105.80. $ 538.45 

5. $160; $208. | 11. 2332.64. (grace). 

6. 1, 12. $371.20. 6. 

Tan0e Vy, 13. 222%. 7 STD: 

8. 81 yd. 14. 32638 m.- 8. 140.064 sq.in. 

9, OA. 9. Nothing. 

10. Bonds .5, %. Art. 379. 10. $285. 

1205-7. ; 11. Rich $800; 

12. 162 sq. ft. cues Foster $1200. 

13. 116 ft. 8} in. - 8 10.50. 12. 336.13+. 

14. 413013: T. 4. $8663. 13. 622, sq. ft. 
Art. 377. ee tbe Art. 382. 

6. 331%. 

oa es 7. $48 gain. 1. 5h. 

2. UTE Ye 8. $ 9774.72 ; 2. $ 18,000. 

3. 22%. $ 64162. 3. $625. 

4. $13.82. 9. $29.75. 4. $115.63. 

5. 173 yd. 10. $212.61. 5. $3010. 

6. 98.81— rd. 11. $6.25. 6. 3125. 

Toate ¥, $817.16. 

8. 19683 sq. in. 8. 42.1+ yd. 

9. 9in. x 9in. Art. 380. 9. $12.10. 

10. $ 140.623. the 164 ite 10. e176. 

11. $ 1871.09. 2. $6150. 11. $26,000. 

12. 162%. 3. $8240. 

18. A $3240; 4. $3761.25. Art. 383. 
B $ 1260; 5. 616.11 + sq. re 
Soh x, 7 ee $38; 

14. 399.999975. 6. 35.99— gal. C $ 26. ; 

15. $48. 752564 bd. ft A aeane 

en Gion é ate 

1. 125%. 10. -75 DO} 6. $152. 

SOt meq tte lt ames 7. $72. 

Sere 12. 8. 1765.17 gal. 

4. $11,880. Art. 381. 9. 9929.4. 

5. $20,000. 1. 13d. more. 10. 12.44 ft. 

G12 4. 2. 64 ft. 1131-710: 

7. $3696. 3. $103.25. 12. $16.80. 

8. 500 men. 4. 53h. 118. 564 sq. ft. 


aa 
= © 


— 
© eo 


10. 
rat 
. 44.74 — 9, 


CO) 00) Sh OF. OE eee 


So Soa 


Art. 384. 


. $33,3331. 


74%. 
$ 266.48. 
50 m. 
12 d. 
$115.20. 
12 rd. 
2 ft. 4 in. 


. 32.725 Ib. 
. $3.3894+. 
. The former by 


$70.30. 


. 64 men. 
<9.0z: 


Art. 385. 


14 bu. 2 
4 qt. 

$ 37.70. 

$ 620.123. 
$ 4504.50. 


pk. 


. $249.64; 


$ 249.52 
(grace). 
2.6— cu. ft. 
as 


$ 55. 


. A $91.30; 


B $114.13; 
C $ 144.57. 
$ 55,000. 

$ 1030.64. 


Art. 386. 


. $72,000. 
. $76. 


5 
C #53 


. $577.40. 


Ls 


SSS So 


$386 ; 600 fr. 


$ 82,187.50. 
116 ft. 

$ 1059.87 + ; 
$ 1059.67 
(grace). 

$ 2125.33!. 


Of B; $1.39. 
159.155 cu. ft. 


Art. 19. 
8472009 ; 
1867.75+ Ib. 
26.4347. 
16.7284 ™, 


1312.359 yd. 


. 395.38 sq. rd. 


1000, 


. $10.63 gain. 


96049, 


. 88.9056, 
. 2,845184400, 

. $3.3108. 

. 5GHa. 138,384, 


264.17 gal. 
800. 


. $3.21. 


196.87549, 


. 7500", 


13.9286947, 


oP @ to 


O DAD MTP ww 


ANSWERS. 
Art. 387. 7. 20.014 m. 
$1.54. 8. 21,3. 
$7.20. 9. $16.80. 
$ 5700. 10. $80,960. 
6. 11. $10,552. 
504 bu. ; 12. $147. 
$ 290.64. 
. 0.02078125. Art. 388. 
1. 38. 


APPENDIX I. 


Art. 21. 

$ 356.40. 
567. 
554.40. 
198.39. 
774.90. 
126.30. 
86.87. 
454.78. 


Art. 23. 


2. $605.02. 


iv) 


OW ADP w 


. $1175.77. 


Art. 25. 


3 mo. 10d. 
1 mo. 24 d. 
2) ADT. 20, 


July 26. 
June 11. 
Dec. 21. 


10. 


11; 
12. 


CS race 


OID Tw ow 


$ 843.36 ; 
Ta do 
Aug. 20. 
Jan. 4. 
Aug. 23. 


Art. 26. 
May 31. 
Oct. 24. 
May 25. 
Nov. 13. 
Jan. 5. 


Art. 28. 
23. 
32. 
36. 
42, 
47, 


aha 


G0 Se St 


23 


$ 2007.50. 

$ 2.912 ; 

$ 912.912. 

$ 9956.86—. 

$ 1337. 
Makes $6.50. 
$ 3.54, 

61% 


i AD ing 
a Lb. 1-4 in; 


156. 


Ceo bole colar wip * 
. . ° . (nd 


— 
on 


te 
. 


STO lt HI 


. 662 sq. ft. 
24 ft: 


— 
° 


(or 


CO Oe Can le 


Mf 


apt er 


25. 

50. 

74 bbl. 

$100. 

8. 

$ 1000. 

20. 

2 —— 0. 

6. 

Li Ep ee ka 
9 x 10, 


Art. 9. 


a — (30 > 3d). 


_ 20 x8 
2 
150 


~~ 100 = 25: 
_ _ 160 
Ui ——= 


3 
r= i — 2. 


a= x t of 189 +2, 


e200 ae 
400” 
or 7 = 400 2: 


==) DLS SA 


or eee 
144 


Art. 10. 
m(a +b). 


ANSWERS. 


APPENDIX IL 


mt 


ie} 
SY & 
~ 
~ 


caotanPr ww 
[on 
wee 


ry 
oO 
— 
= 


Art. 16. 


12. 

12: 

24, 

$ 300. 

. $56. 
36m. 
120. 

$ 31.8852. 
$ 84. 


OOD AMP ww 


_ 
od 


Axt. “17. 


SHOHMPABDARwD 


se 
Loe 
wo 
(<2) 


144. 
12: 
4, 

Art. 18. 
$210; $70. 
20 qt. ; 25 qt. 
9714 
48; 39. 
$0.40; $080. 
A 2180; B 800; 


C 320. 


168. 9. 150. 
60. 10a 
Art. 19. 


$ 10,000 ; $5000. 
12. 


3. F $94; G $122; 


Oe 


H $56. 

200 gal. each. 
132 

8. 

Lis entlo: 
84; 91; 98. 
60 ; 20. 

$ 871. 


Art. 20. 
0. 
10 x7. 
7(b+ 6). 
— 12%. 


Arteais 
SET ViDy 
1m. 
24 rece’d. 


Lb. 


. @bex?. 


be Gert 1) + 2. 


: a+(b-—c)-e. 


x—(y—z2)+a.- 


Art. 25. 


10 abe. 

18 abe. 

36 axy. 
48 bey. 
228 abxy. 
63 bedz. 
2(abcx)?. 
4 atc?mat. 
8 ab. 

6 abcrny. 
a. 

(m + n)?. 


6(a — Db). 
6(a — b)?. 


(m+ n)?. 
12 (“+ y)?. 
343+ 3ay? + 3 az. 


. yz 4+ ay?z + yz. 
. Gam. 


10 axty? + Ls ar ey? + | 
20 ax? y?z. ) 
14a+21b+7e. | 
(Combine _ before 
multiplying. ) 


10. 


1 


2. 


3. 


aYZ + Ak2Z + ALY 


ANSWERS. 


Art. 29, 


Art. 30. 


lbay, be. 
10 a2’ 10 ab 
3%, 2am. 
6a2y’ 6 ay 
a Dae ae 

48 ab2m2’ 
3 b?m?z 
48 ab?m? 


Samy 
48 ab?m? ’ 


Art. 31. 


4a+2b? 


xy 


or 


LYZ 


aye + £2 + LY) | 


LYZ 


22m — 3am 


nin 


| 


25 


2 wyz? +12 22+ 16 oy? 


8 ey?Z2 


4¢c— 2ad* 


cdy 


3y(a—b)+3a(a+d) 


rY 
adn + ben — bdm. 
bdn 


a(emy + dn) | 


a(b? + x)—y, 


ad 


ite 

4 ay 

2@mnxy , 
3 azz 


" 2bced 


(a+ b)\@+Y)- 


Art. 33. 


bo 
a 


xn & 
bo 


5 m? 


<~ ar | 


2 


— 


arb 


oO Xe 


2(2a + b*) 


K(2a + b)ay? 


1¢ca 


" Dbz 


26 


> OO 3 


10. 


3.ab 

aly + da) 
3 cd3y 

Ded 

ab? 


Art. 34. 
Beit: 
S53. 
15; 4. 


$75; $100. | 18. 


—_ : 
FOO OA ar & Ww 


ANSWERS. 


bs enee 


2 

. 30; 24,° 3 
18; 12. 4. 

= Alea es 5 


LS: 


27; 74. 


Art. 36. 


A $50; 10. 


B $40. 


. $550; $300. 
. 64; 36. 


40; 24, 


. Tea ib¢: 


Coffee 40 ¢. 
70; 30. 
63 ; 27. 
20; 12. 

A $270; 
B $252. 


4 
9° 


at 


me 
& 


oh 
aye 


ik ; 


ee 


bag 
Ai de eat hi 


fh iy : 


aoe 


7 


= 
Tele 
a 
eh. 


er a 
~ a er 
eet te — — “ 


<3 


eacstat Seater} 3; 


Serbeteittstmes 


errs etree 


Sette 
Sorse-sesesses 


Setpetriieciese 


ors 


tissseseciss 


So BS ye p+ 
wey eeegtse pasesaee 


af 
stectess 


ia 


Baas 


Esser 


UNIVERSITY OF ILLINOIS-URBANA 
513S08E C001 V002 


THE ESSENTIALS OF ARITHMETIC ORAL AND WR 


TN 


3.0112 0171 


streSete Syed e7 


SOI 


